david ronis mcgill universityronispc.chem.mcgill.ca/ronis/chem593/course_pac.pdf · statistical...

Josiah Willard GibbsLudwig Boltzmann

Albert Einstein

John Gamble Kirkwood

Chemistry 593:

Statistical Mechanics

David Ronis

McGill University

Lars Onsager

Robert W. Zwanzig

Irwin Oppenheim Ryogo Kubo

Winter, 2018

Chemistry 593: Statistical Mechanics

David Ronis

© 2018

McGill University

All rights reserved. In accordance with Canadian Copyright Law, reproduction of this material, in whole or in part,

without the prior written consent the author is strictly prohibitied.

Last modified on 18 April 2018

Winter, 2018

Chemistry 593 -2- Problem Set 4

Table of Contents

1. General Information . . . . . . . . . . . . . . . . . . . . . . 3

2. Stirling´s Formula . . . . . . . . . . . . . . . . . . . . . . 6

3. Ideal Bose & Fermi Gases . . . . . . . . . . . . . . . . . . . . 9

3.1. Introduction and General Considerations . . . . . . . . . . . . . . . 9

3.2. Results for Particles in a Cubic Box . . . . . . . . . . . . . . . . 12

3.2.1. The Case For Boltzmann Statistics . . . . . . . . . . . . . . . . 15

3.3. Results Specific to Fermions or Bosons . . . . . . . . . . . . . . . 18

4. The Semi-Classical Limit . . . . . . . . . . . . . . . . . . . . 26

5. Dense Gases: Virial Coefficients . . . . . . . . . . . . . . . . . . 32

6. Correlation Functions and the Pressure . . . . . . . . . . . . . . . . 40

7. Normal Mode Analysis . . . . . . . . . . . . . . . . . . . . . 43

7.1. Quantum Mechanical Treatment . . . . . . . . . . . . . . . . . 43

7.2. Normal Modes in Classical Mechanics . . . . . . . . . . . . . . . 45

7.3. Force Constant Calculations . . . . . . . . . . . . . . . . . . . 45

7.4. Normal Modes in Crystals . . . . . . . . . . . . . . . . . . . 47

7.5. Normal Modes in Crystals: An Example . . . . . . . . . . . . . . . 49

8. The q → 0 Limit of the Structure Factor . . . . . . . . . . . . . . . 54

9. Gaussian Coil Elastic Scattering . . . . . . . . . . . . . . . . . . 56

10. Some Properties of the Master Equation . . . . . . . . . . . . . . . 58

11. Critical Phenomena . . . . . . . . . . . . . . . . . . . . . . 60

12. Liouville´s Equation . . . . . . . . . . . . . . . . . . . . . 83

13. Nonequilibrium Systems . . . . . . . . . . . . . . . . . . . . 89

14. Langevin Equations . . . . . . . . . . . . . . . . . . . . . 106

15. Projection Operators . . . . . . . . . . . . . . . . . . . . . 109

16. Inelastic Light Scattering . . . . . . . . . . . . . . . . . . . . 113

17. Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . 127

17.1. Problem Set 1 . . . . . . . . . . . . . . . . . . . . . . . 127

17.2. Problem Set 2 . . . . . . . . . . . . . . . . . . . . . . . 129

17.3. Problem Set 3 . . . . . . . . . . . . . . . . . . . . . . . 131

17.4. Problem Set 4 . . . . . . . . . . . . . . . . . . . . . . . 132

General Information -3- Chemistry 593

1. General Information

Professor David Ronis Otto Maass, Room 426

Lectures: MWR 1:35-2:25 P.M. in BURN 1B24

Course Website: https://ronispc.chem.mcgill.ca/ronis/chem593

Case sensitive username and password needed for total acccess.

Username: chem593 Password: Boltzmann

Chemistry 593: Statistical Mechanics

Course Description

Intermediate topics in statistical mechanics, including: modern and classical theo-ries of real gases and liquids, critical phenomena and the renormalization group,time-dependent phenomena, linear response and fluctuations, inelastic scattering,Monte Carlo and molecular dynamics methods.

1.1. TEXT

D. A. McQuarrie, Statistical Mechanics.

1.1.1. SUPPLEMENTARY TEXTS

1. L.K. Nash, Elements of Statistical Thermodynamics (Introductory)

2. T. L. Hill, An Introduction to Statistical Thermodynamics (Dover Publications, intro-ductory, cheap) D. A. McQuarrie, Statistical Mechanics.

3. T. L. Hill, Statistical Mechanics. (Dover, advanced, slightly old fashioned)

4. K. Huang, Statistical Mechanics. (Advanced).

5. L. E. Reichl, A Modern Course in Statistical Physics. (Intermediate).

1.1.2. General References

Old Classics

J. W. Gibbs, Elementary Principles in Statistical Mechanics

R. C. Tolman, The Principles of Statistical Mechanics

H. S. Green, The Molecular Theory of Liquids

J. Frenkel, Kinetic Theory of Liquids

Advanced References

Winter, 2018

Chemistry 593 -4- General Information

A. Munster, Statistical Thermodynamics Volumes I and II (encyclopedic)

Landau and Lifshitz, Statistical Physics

R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics

Hirschfelder, Curtiss and Bird, Molecular Theory of Gases and Liquids

The Liquid State

Hansen and MacDonald, Theory of Simple Liquids

Frisch and Lebowitz, Classical Liquids

Critical Phenomena

S. K. Ma, Modern Theory of Critical Phenomena

H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena

Time Dependent Phenomena

P. Resibois and M. DeLeener, Classical Kinetic Theory of Fluids

D. H. Zubarev, Nonequilibrium Statistical Mechanics

S. Chapman and T. Cowling, The Mathematical Theory of Nonuniform Gases

D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions

1.2. Grading Scheme

The grade in this course will be computed as follows:

Grading Scheme: CHEM 593

Homework 10%Midterm 40%Term Paper 50%

Note:

McGill University values academic integrity. Therefore all students must understand the

meaning and consequences of cheating, plagiarism and other academic offenses under the

code of student conduct and disciplinary procedures (see www.mcgill.ca/integrity for more

information).

In accord with McGill University’s Charter of Students’ Rights, students in this course

have the right to submit in English or in French any written work that is to be graded.

In the event of extraordinary circumstances beyond the University’s control, the content

and/or evaluation scheme in this course is subject to change.

Winter, 2018

General Information -5- Chemistry 593

1.3. Course Outline

Outline: Chemistry 593, Winter, 2018

Lecture Topic

Lecture 1 Quantum corrections & the classical limitLecture 2 "Lecture 3 Introduction to dense gasesLecture 4 The 2nd virial coefficientLecture 5 Introduction to reduced distribution functionsLecture 6 Reduced distribution functions and thermodynamicsLecture 7 "Lecture 8 "Lecture 9 Elastic scattering and the structure factorLecture 10 Theories of reduced distribution functions I. YBG and Kirkwood equations and the superposition

approximationLecture 11 "Lecture 12 "Lecture 13 Theories of reduced distribution functions II: The direct correlation function, Ornstein-Zernike

equation, and HNC, PY and MSA closures.Lecture 14 "Lecture 15 "Lecture 16 Monte Carlo methodsLecture 17 "Lecture 18 Perturbation theory of liquidsLecture 19 WCA vs BH (Canada/USA Round I).Lecture 20 Introduction to critical phenomena: Landau-Ginzberg and Weiss Mean field theories, critical expo-

nents and universality.Lecture 21 "Lecture 22 "Lecture 23 Scaling and renormalization group: Survey of experimental results, the Rushbrooke inequality,

Widom’s scaling hypothesis, Kadanoff’s renormalization group, fixed points and scaling, real spacerenormalization method.

Lecture 24 "Lecture 25 "Lecture 26 "Lecture 27 Time dependent phenomena: Brownian motion and Langiven equations, classical and quantum

Liouville Theorem, Langevin equations, linear response theory: susceptibilities and response func-tions, fluctuation dissipation theorem, projection operators and separation of time scales, hydrody-namic equations and Green-Kubo formulas.

Lecture 28 "Lecture 29 "Lecture 30 "Lecture 31 "Lecture 32 "Lecture 33 Molecular dynamics methodsLecture 34 "Lecture 35 Inelastic light scattering: fluctuations and scattering, detection methods, optical filters, optical mix-

ing: heterodyne and homodyne methods, the Ralyeigh-Brillouin spectrum.Lecture 36 "Lecture 37 "Lecture 38 "Lecture 39 "

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Chemistry 593 -6- Stirling´s Formula

2. Stirling´s Formula

There is a simple way to obtain an asymptotic expansion for the factorial, n!, for large argu-ments. First define a new function, called the gamma-function, as

Γ(x + 1) ≡ x! ≡ ∫∞0

ds sxe−s = x x+1 ∫∞0

ds exp x[ln(s) − s], (2.1)

where the last equality follows by making the change of variables s → xs. The function Γ(x) iscalled the gamma function. It is easy to show, either by integrating by parts or by noting thatΓ(x + 1) = xΓ(x), that x! = n! when x = n, an integer.

The integrand appearing in Eq. (2.1) is shown in the following figure.

Fig. 2.1. The integrand in Eq. (2.1).

When x is large, the integrand is large and sharply peaked around the maximum. In this case, itis profitable to Taylor expand the integrand around the maximum of the argument to the expo-nential, i.e., about s = 1 in Eq. (2.1). Hence, we write

x[ln(s) − s] = −x − x∞

n=2Σ (−1)n(s − 1)n

n. (2.2)

If we only keep the n = 2 term in the sum in Eq. (2.2), it follows that the integrand in Eq. (2.1)decays like a Gaussian centered at s = 1, with width x−1/2. Remembering that x is large, we cannow extend the lower limit of integration in Eq. (2.1) to −∞ with exponentially small error.Finally, we make the change of variable s → 1 + ∆/x1/2 and rewrite Eq. (2.1) as

x! = x x+1/2e−x ∫∞−∞

d∆ exp−

1

2

∆2 −∞

n=3Σ (−1)n∆n

nx(n−2) / 2

. (2.3)

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Stirling´s Formula -7- Chemistry 593

The terms in the sum are at least as small as x−1/2, and hence, to leading order can bedropped. The resulting integral is easily done, and hence,

x!∼(2π )1/2 x x+1/2e−x , (2.4a)

which implies that

ln(x!) ∼ x ln(x) − x + ln(2π x)1/2

. (2.4b)

Clearly, for x∼O(1023), the last logarithm is negligible, and the simple form of Stirling’s formulaln(x!)∼x[ln(x) − 1], is obtained. A comparison of the approximate and exact results is shown inFig. 2.2.

Fig. 2.2. Comparing Stirling’s approximations to ln(x!). The insetshows smaller x.

Finally, some of the numerical results used to make Fig. 2.2 are shown in the following table:

Table 2.1: Testing Stirling’s Approximation

ln(x!) x[ln(x) − 1] x[ln(x) − 1] + ln[(2π x)1/2](exact) Value ∆% Value ∆%

x

1 0 -1 ∞ -0.0810615 ∞5 4.78749 3.04719 36.4 4.77085 0.348

10 15.1044 13.0259 13.8 15.0961 0.055250 148.478 145.601 1.94 148.476 1.12×10−3

100 363.739 360.517 0.886 363.739 2.29×10−4

500 2611.33 2607.3 0.154 2611.33 6.38×10−6

1000 5912.13 5907.76 0.074 5912.13 1.41×10−6

With a little extra effort you can work out the next term in the expansion, by Taylor expand-ing the exponential of the sum in Eq. (2.3) and keeping the next to leading order terms which

Winter, 2018

Chemistry 593 -8- Stirling´s Formula

give nonvanishing integrals. If you were to carry out this procedure to arbitrarily large order,would you expect the resulting series to converge? Why?

Winter, 2018

Ideal Bose & Fermi Gases -9- Chemistry 593

3. Ideal Bose & Fermi Gases

3.1. Introduction and General Considerations

When we consider quantum statistical mechanics of identical particles, we abandon theconventional, classical, notion of states (i.e., as lists of quantum numbers assigned to the parti-cles) and instead focus on "occupation numbers", ni , the number of particles in 1-particle state i

(e.g., a single hydrogenic electron or a single particle in a box). Indistinguishably implies thatthe quantum probability density, |ψ (1, 2, 3, 4, . . .)|2, be symmetric under exchange of any particlelabels, and, in turn, this means that ψ (1, 2, . . .) = ± ψ (2, 1, . . .). Quantum field theory and specialrelativity show that the + sign is used for integer spin particles, generically called Bosons, whilethe − sign is used for half-odd integer spins, referred to as Fermions.

Tw o see how this works, consider the following 2-particle example: each particle can be inone of two states, |α , i > or |β , i >, i = 1, 2, with energies εα and ε β , respectively. If we ignorethe consequences of indistinguishability we can construct 4 possible states for this system asshown in the following table:

Table 3.1: Possible States and Energies for a Two Lev el System with Two Particles

State Wav efunction Energy nα nβ Particle

1 |α , 1 > |α , 2 > 2εα 2 0 Bosons2 2−1/2(|α , 1 > |β , 2 > + |β , 1 > |α , 2 >) εα + ε β 1 1 Bosons3 2−1/2(|α , 1 > |β , 2 > − |β , 1 > |α , 2 >) εα + ε β 1 1 Fermions4 |β , 1 > |β , 2 > 2ε β 0 2 Bosons

Q(N = 2, T , V ) =

e−εα /kBT + e−ε β /kBT

2

, for distinguishable particles (4 states)

e−(εα +ε β )/kBT , for Fermions (1 state)

e−2εα /kBT + e−(εα +ε β )/kBT + e−2ε β /kBT , for Bosons (3 states).

(3.1)

For systems with more than two particles we simply symmetrize or anti-symmetrize the wav e-function (i.e, add all possible permutations of the particle labels together) for Bosons or use theso-called Slater determinant for Fermions.1 This shows that ni = 0, 1, 2, . . . for Bosons, while

1For Bosons this means

Ψ(1, 2, . . . , N ) =Permutations

Σ P[ψ a(1) . . .ψ z(N )],

where, P permutes the particle labels and where ψ a(1) is an orbital for one particle in 1-particle state a

(e.g., for atoms, an atomic orbital). For Fermions we construct the so-called Slater determinant, i.e.,

Ψ(1, 2, . . . , N ) =

ψ a(1)

ψ b(1).

.

.ψ z(1)

ψ a(2)

ψ b(2).

.

.ψ z(2)

. . .

. . .

. . .

. . .

. . .

ψ a(N )

ψ b(N ).

.

.ψ z(N )

.

Chemistry 593 -10- Ideal Bose & Fermi Gases

ni = 0, 1 for Fermions, as required by the Pauli Exclusion Principle.1 In either case, the wav e-function is determined, up to an overall phase factor, by specifying the ni’s.

Since E = Σi niε i , it follows that the canonical partition function is

Q(N , V , T ) =

iΣ ni = N

1 or ∞

nα = 0Σ

1 or ∞

nβ = 0Σ . . . exp

−

iΣ ε i ni /kBT

, (3.2)

where the upper limits of the sums depend on whether Fermions or Bosons are under considera-tion. The sums aren’t easy to do when constrained to have Σi ni = N . Since N fluctuates in thegrand canonical ensemble, we expect that the constraint on N to disappear. Moreover, we’veshown that the choice of ensemble does not matter when calculating thermodynamic quantities.Hence, we switch to the grand canonical ensemble, with partition function

Ξ =∞

N = 0Σ eβ µN

iΣ ni = N

1 or ∞

n1 = 0Σ

1 or ∞

n2 = 0Σ . . . e

−βiΣ ε i ni

=1 or ∞

n1 = 0Σ

1 or ∞

n2 = 0Σ . . . e

−βiΣ(ε i−µ)ni

=i

Π

1 or ∞

ni = 0Σ eβ (µ−ε i)ni

=i

Π1 ± eβ (µ−ε i)

±1

, (3.3)

where we’ve gone back to our earlier notation of naming states by a roman index, β and µ havetheir usual meanings, and where the last equality follows by explicitly carrying out the sums forFermions (upper signs) or Bosons (lower signs).

Once we have the grand canonical partition function, thermodynamic functions areobtained in the usual manner, e.g.,

β pV = ln(Ξ) = ±iΣ ln

1 ± eβ (µ−ε i)

, (3.4)

⟨N ⟩ =

∂ ln(Ξ)

∂β µ

T ,V

=iΣ ⟨ni⟩ =

iΣ 1

eβ (ε i−µ) ± 1, (3.5)

and

⟨E⟩ =

∂ ln(Ξ)

∂ − ββ µ,V

=iΣ ⟨ni⟩ε i , (3.6)

where the average occupation numbers are

Note that neither expression is normalized. The Slater determinant is odd under exchange of any two rowsand vanishes if two or more rows are the same, thereby having the required symmetry and Pauli exclusionprinciple.

Winter, 2018


⟨ni⟩ =1

eβ (ε i−µ) ± 1, (3.7)

and are known as the Fermi-Dirac (+ sign) or Bose-Einstein (- sign) distributions.

Finally consider the heat capacity. From Eqs. (3.6) and (3.7) it follows that

CV =kB

2

i, jΣ ⟨n j⟩

2eβ (ε j−µ)⟨ni⟩

2eβ (ε i−µ)[β (ε i − ε j)]

2

jΣ ⟨n j⟩

2eβ (ε j−µ)

, (3.8)

where the details of calculation are given in Appendix A. Note that CV is positive, as expected.

The quantity ⟨ni⟩2 exp[β (ε i − µ)] plays a key role, and is shown in Figs. 3.1 and 3.2, for Fermions

and Bosons, respectively. An approximate expression for Fermions for 4⟨ni⟩2ex can be obtained

by noting that the Taylor expansion of ln(4⟨ni⟩2ex)∼ − x2/4 + O(x4), where x ≡ β (ε i − µ); hence,

4⟨ni⟩2ex ≈ e−x2/4

Fig. 3.1. Mean occupation numbers, ⟨ni⟩, and 4⟨ni⟩2eβ (ε −µ) for

Fermions. The spin degeneracy is not included. Notice howthe orbitals change from filled to empty over a range ofβ (ε − µ) ∼ ± 5 or, for T = 300K , over a range∆(ε − µ) ∼ ± 2. 5kBT = ±0. 065 ev. The plot of ⟨ni⟩

2eβ (ε −µ)

shows that the main contributions to the heat capacity comefrom a narrow band of 1-particle states near the Fermi energy.Finally, the dotted brown line is the approximation to4⟨ni⟩

2eβ (ε −µ) discussed in the text.

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Fig. 3.2. Mean occupation numbers, ⟨ni⟩, and ⟨ni⟩2eβ (ε −µ) for

Bosons. The spin degeneracy is not included.

Equation (3.8) leads to an approximate expression for CV . Appendix A shows a more rig-orous approach2

3.2. Results for Particles in a Cubic Box

In order to proceed we need a concrete model for the 1-particle states and energies. Thesimplest is probably the particle with mass m in a cubical box of side length L. This has ener-gies:

ε nx ,ny,nz=

h2

8mL2(n2

x + n2y + n2

z), where ni = 1, 2, 3, . . . . (3.9)

Note that the ground state energy, ε1,1,1 = 3h2/(8mL2) → 0 as L → ∞. When the sum over sin-gle particle states (i.e., over positive ni , i = x, y, z,) is approximated as an integral and the resulttransformed to polar coordinates, we find that all the examples at the end of the previous sectioncan be written as

(2S + 1)π2 ∫

∞0

dn n2 f (λ , β ε n), (3.10)

where the factor of (2S + 1) accounts for the spin degeneracy of a particle with spin S, the factorof π /2 = 4π /8 is the area of the unit sphere in the octant where ni > 0, i = x, y, z, and λ ≡ eβ µ is

the proper activity. Next, change variables by letting n = √ 8m/h2 Lε 1/2 and rewrite Eq. (3.10) as

∫∞0

dε g(ε ) f (λ , β ε ), (3.11)

2See J.E. Mayer and M.G Mayer, Statistical Mechanics, (John Wiley & Sons, Inc., 1940), Sec. 16g.

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where

g(ε ) ≡ 2π (2S + 1)

2m

h2

3/2

V ε 1/2 (3.12)

is known as the density of states and can be interpreted as the number of states per unit energywith energies between ε and ε + dε . With this, the general results given at the end of the pre-ceding section become

β pV = ± ∫∞0

dε g(ε ) ln1 ± λe−β ε

, (3.13a)

⟨N ⟩ = ∫∞0

dε g(ε )λe−β ε

1 ± λe−β ε , (3.13b)

and

⟨E⟩ = ∫∞0

dε g(ε )ελe−β ε

1 ± λe−β ε . (3.13c)

Note that Eqs. (3.13b) and (3.13c) can be obtained by taking the usual derivatives of β pV , i.e.,ln(Ξ), cf. Eq. (3.13a).

Next we expand the integrands into Taylor series in λ , specifically,

± ln1 ± λe−β ε

= −+

∞

j = 1Σ (−+ λ) j

je− jβ ε (3.14a)

and

λe−β ε

1 ± λe−β ε = ±

∂ ln(1 ± λe−β ε )

∂β µ

β ,V ,N

= −+∞

j = 1Σ (−+ λ) je− jβ ε , (3.14b)

and change variables yet again by letting ε ≡ (kBT / j)x . This gives:

β pV = −+2

π 1/2

V

Λ3(2S + 1)

∞

j = 1Σ (−+ λ) j

j5/2Γ(3 / 2), (3.15a)

⟨N ⟩ = −+2

π 1/2

V

Λ3(2S + 1)

∞

j = 1Σ (−+ λ) j

j3/2Γ(3 / 2), (3.15b)

and

⟨E⟩ = −+2

π 1/2

V

Λ3kBT (2S + 1)

∞

j = 1Σ (−+ λ) j

j5/2Γ(5 / 3), (3.15c)

Winter, 2018


where Λ ≡ h/√ 2π mkBT is the thermal de Broglie wav elength, and have expressed the remainingintegrals as Γ-functions,3 specifically, Γ(3

2) = 1

2π 1/2, for β pV and ⟨N ⟩, and Γ(5

2) = 3

4π 1/2 for ⟨E⟩.

When these are used in Eqs. (3.15a)−(3.15c) and the results rearranged, we find that

ρΛ3 = (2S + 1)G3/2(λ) (3.16a)

and

β pΛ3 =2β ⟨E⟩Λ3

3V= (2S + 1)G5/2(λ), (3.16b)

where

Gz(λ) ≡1

Γ(z) ∫∞0

dx x z−1 λe−x

1 ± λe−x= λΦ(−+ λ , z, 1) = −+

∞

j=1Σ j−z(−+ λ) j , (3.16c)

generalizes Hill’s Fz(α ),4 the function Φ(a, b, c) is discussed in Gradshetyn and Ryzhik,5 andρ = ⟨N ⟩/V is the number density. The first equality in Eq. (3.16b) holds for the individual statesfor the particle in a box, i.e., pn = 2ε n/3V .

These sums converge absolutely for |λ | < 1 (µ < 0) and can be summed numerically,although the effort to compute the integrals numerically is comparable and must be used forFermions for λ > 1. Some results are shown in Fig. 3.3. When λ = 1 (i.e., µ = 0), the sums inEq. (3.16c) reduce to the Riemann zeta functions as shown in the following table:

Table 3.2: Connections to the Riemann- Zeta FunctionsGz(1): Formulas and Numerical Values

z Fermions: (1 − 21−z)ζ (z) Bosons: ζ (z)

3/2 0.7651 2.6125/2 0.8671 1.341

3One of the ways of defining the Γ function is via its integral representation

Γ(z) ≡ ∫∞

0dx x z−1e−x ,

which includes all of the remaining integrals. For more properties of these functions, see, e.g., the Hand-

book of Mathematical Functions, tenth edition, M. Abramowitz and I.S. Stegun, eds., ch. 6.4T. L. Hill, An Introduction to Statistical Thermodynamics (Dover Publications), p. 418.5I.S. Gradshetyn and I.M. Ryzhik, Table of Integrals, Series and Products (Academic Press. New York,1980), Sec. 9.55.

Winter, 2018


Fig. 3.3. Proper activities, λ , and equations of state for ideal Boltz-mann, Fermi-Dirac, and Bose-Einstein gasses. The spin degeneracyfactor has not been included. Note that the figure was actually obtainedby choosing a range of λ’s, calculating ρΛ3, and transposing the plot.

The first thing to notice is that the curves are universal, in that all ideal systems’ properties,once corrections for the spin degeneracy are made, will fall on the same curves. Also notice thepositive (Fermions) and negative deviations (Bosons) from ideal gas behavior. For Fermions,these positive deviations arise because the Pauli exclusion principle which disallows two non-interacting particles to occupy the same state or space, and thus has an effect similar to stericrepulsion). Bosons don’t hav e to obey the exclusion principle and can have any number of parti-cles in the same state, thereby acting like an attraction.

3.2.1. The Case For Boltzmann Statistics

It turns out that there is a simple fix that allows us to treat the particles as if they were dis-tinguishable. Consider a separable Hamiltonian which has 1-particle energy levels, ε i = i, fori = 1, 2, 3, . . .. Let’s examine the states for a system comprised of three identical particles, whereHΨ(a, b, c) = 9Ψ(a, b, c)

Winter, 2018


Table 3.3: States where HΨ(a, b, c) = 9Ψ(a, b, c)

Degeneracy

ΩFD ΩDistinguishable ΩBE

1-Particle States Nonzero ni’s

(7, 1, 1) n7 = 1 and n1 = 2 0 3 1(6, 2, 1) n6 = 1, n2 = 1, and n1 = 1 1 3! 1(5, 3, 1) n5 = 1, n3 = 1 and n1 = 1 1 3! 1(5, 2, 2) n5 = 1 and n2 = 2 0 3 1(4, 4, 1) n4 = 2 and n1 = 1 0 3 1(4, 3, 2) n4 = 1, n3 = 1, and n2 = 1 1 3! 1(3, 3, 3) n3 = 3 0 3! 1

The degeneracy for Fermions, ΩFD, vanishes whenever the Pauli exclusion principle is violated,i.e., whenever ni > 1, as expected. Since the wav efunction for Bosons can always be sym-metrized, any choice of the ni’s giv es a single wav efunction, and ΩBE = 1. Finally, the degenera-cies for the distinguishable cases have 0 < ΩDistinguishable ≤ N !, the equality holding when all the1-particle states are different. This discussion can be summarized by noting that

ΩFD ≤ΩDistinguishable

N !≤ ΩBE ,

where equality holds when all the 1-particle states are different.

If the number of available states is much larger than the number of particles, then most ofthe states included in the partition function obey the Pauli exclusion principle, and the distin-guishable calculation over-counts these by the same factor of N !. Hence, a corrected partitionfunction becomes

QBoltzmann =QDistinguishable

N !.

This approach is known as Boltzmann statistics. Note that each component of a mixture contrib-utes a N ! factor.

Returning to the particle in a box example, we now hav e

QBoltzmann =(V Λ−3)N

N !

and thus

β A = − log(Q) = −N log(V Λ−3) + N [log(N ) − 1] = N [log(ρΛ3) − 1],

where we have used Stirling’s formula and ρ ≡ N /V is the number density.6 The Gibbs paradox

is resolved!

The ideal gas model makes it fairly straightforward to quantify the condition necessary forBoltzmann statistics to be used, namely, that the number of accessible states, NStates, greatlyexceeds the number of particles, N . As we hav e just shown, when this holds, the resolution of

6Hill writes this as a single logarithm as N log(ρΛ3/e).

Winter, 2018


Gibbs’ paradox is to simply divide the partition function, calculated as if the particles were dis-tinguishable, by N !. The argument is simple and is essentially that presented in Hill.7 To begin,note that the particle in a box quantum numbers, ni , i = x, y, z, which when plotted, form a threedimensional cubic lattice in the positive octant, each lattice point corresponding to a state. Interms of the lattice cells, each has unit volume and each corresponds to a single state (to be sure,there are eight lattice points per cell, but each is shared with 8 neighbors), Thus, if we ignoreedge effects, the number of states with energies less than εmax is just the volume of of an eighthof a sphere with radius, n, less than the n corresponding to the εmax. By using Eq. (3.9), the cut-off radius is n = (8mεmax/h2)1/2 L, and the corresponding number of states is simply

NStates ≡1

8

4π n3

3=

4π3

2mεmax

h2

3/2

V .

By taking εmax = 3kBT /2 and requiring that NStates >> N , we see that

ρΛ3 <<

6

π

1/2

(3.17)

in order to use Boltzmann statistics. Perhaps more physically, the number of particles per vol-ume corresponding a size comparable to the thermal de Broglie wav elength is small, and hence,the number of particles within a thermal de Broglie wav elength, the scale at which quantumeffects (e.g., wav e-particle duality) become important, is small. Note that raising temperatureand/or mass, and/or lowering the density favor the use of Boltzmann statistics.

Fig. 3.4. The temperature dependence of the thermal deBroglie wav elength for various gases. Note that with theexception of the electron, Λde Broglie is comparable to orsmaller than the size of an atom at room temperature.

7T.L. Hill, op. cit., Sec. 4.1.

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Some specific results are given in the following table:8

Table 3.4: Is Boltzmann Statistics Valid?

Element (State) T (K) (π /6)1/2 ρΛ3

He (l) 4 1.6He (g) 4 0.11He (g) 20 2×10−3

He (g) 100 3.5×10−5

Ne (l) 27 1.1×10−2

Ne (g) 27 8.2×10−5

Ne (g) 100 3.1×10−6

Ar (l) 86 5.1×10−4

Ar (g) 86 1.6×10−8

Kr (l) 127 5.4×10−5

Kr (g) 127 2.0×10−7

300 1465electrons inmetals (Na)

Thus, we see that except for ultra-low temperatures and the lightest elements (both probablyaren’t of much interest to chemists), Boltzmann Statistics should be an excellent approximation.There is one major exception, namely electrons around room temperature, something that hasmajor ramifications for bonding, etc..

3.3. Results Specific to Fermions or Bosons

3.3.1. Fermions

As we saw in the preceding section, cf. Fig. 3.1, the Fermi-Dirac distribution is basicallyflat for ε < µF and falls off rapidly for ε > µF , where µF is the chemical potential, known as theFermi energy to honor Enrico Fermi; hence, a reasonable approximation is to cut off the integra-tions at ε = µF and let ⟨ni⟩∼2S + 1 for ε < µF . For example, by using Eqs. (3.13b) and (3.13c) itfollows that

⟨N ⟩ ≈ ∫µF

0dε g(ε ) =

4π (2S + 1)

3

2mµF

h2

3/2

V (3.18)

or

µF ≈h2

8m

6ρπ (2S + 1)

2/3

. (3.19)

Similarly,

8From, D.A. McQuarrie, Statistical Mechanics, (Harper and Row Pub., Inc., New York, NY, 1973), Table4-1, p. 72.

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⟨E⟩ ≈ ∫µF

0dε g(ε )ε =

4π (2S + 1)

5

2m

h2

3/2

V µ5/2F =

3

5⟨N ⟩µF . (3.20)

As the temperature is reduced, the approximations we’ve just made become more accurate;hence, the simple results just obtained become more valid and represent the ground state configu-ration of the Fermi (ideal) gas. Finally, the approximations leading these results don’t lend them-selves well to quantities like like the heat capacity. As we saw above, CV , arises from a smallband of energy levels where ε ≈ µF ; these make CV nonzero and its calculation more compli-cated.

By using Eq. (3.8) and the approximate expression for ⟨ni⟩eβ (ε i−µ) it follows that

CV

kB

∼ kBT

8

∫∞−∞

dx g(kBTx + µ)e−x2/4 ∫∞−∞

dy g(kBTy + µ)e−y2/4(x − y)2

∫∞−∞

dx g(kBTx + µ)e−x2/4

∼ kBTg(µ)

4 ∫∞−∞

dx e−x2/4 x2 = kBTg(µ)π 1/2 = 2(2S + 1)

2π mkBT

h2

3/2

V

µ

kBT

1/2

,

where the last expression was obtained by using Eq. (3.12). By using Eq. (3.18) to eliminate thevolume in favor of ⟨N ⟩ it follows that

CV

⟨N ⟩kB

∼ π 1/23kBT

2µF

=3τ

2π 1/2,

where τ ≡ π kBT /µF is a reduced temperature. The more rigorous calculation given in the Ap-pendix, cf. Eq. (A13), gives π τ /2, i.e., a 46% error. In any event, both calculations show that thatCV ∝T as T → 0, which is consistent with the Third Law of Thermodynamics. Also note thatkBT /µF << 1; e.g., if µF = 1 eV, and T = 300K then CV /kB ≈ 0. 069.

3.3.2. Bosons

The Bose gas thermodynamic properties can be obtained from the results given in Eqs.(3.13a−c) or (3.4−7). In particular, from Eq. (3.13b), i.e.,

⟨N ⟩ = ∫∞0

dεg(ε )

eβ (ε −µ) − 1, (3.21)

noting that the particle in a box model has ε ≥ 0 as L → ∞, it follows that µ < 0 or λ < 1 so thatthe integrand in Eq. (3.21) be positive and integrable. However, from the data shown in Fig. 3.3,it follows that λ = 1 for ρΛ3 ≈ 2. 612. For 4 He, using the experimental density of liquid helium(0. 145g/cm3) this occurs at T = 3. 14K . Experiment shows that there is a transition at 2.18K.(What contributes to the discrepancy?)

In order to resolve this problem, note that the exact number of particles in the ground state(ε = 0) in the grand canonical ensemble is

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(2S + 1)λ

(1 − λ). (3.22)

On the other hand, the factor of g(ε )∝ε 1/2 in Eq. (3.21) shows that the ground state doesn’t con-tribute at all. This makes sense if the states form a continuum; the probability of finding anyexact state (not a narrow band of them) vanishes, and we must introduce probability densities.On the other hand, the ground state contribution to the partition function is the most divergent!

The way out of this seeming paradox is to postulate what is known as the two fluid model.We write

⟨N ⟩2S + 1

=λ

1 − λ+ 2π

2m

h2

3/2

V ∫∞0

dεε 1/2

eβ ε − 1(3.23a)

=λ

1 − λ+

V

Λ3ζ (3 / 2), (3.23b)

where the first term is the number of particles in the ground state, the so called Bose condensate,while the second accounts for those in excited states. Finally, if we assume that µ → 0 −, andexpand λ in the denominator of the first term in Eq. (3.23b) we find that

β µ ∼ −Λ3

V

ρΛ3 − ζ (3 / 2)

, (3.24)

where we have taken S = 0 with ρΛ3 ≥ ζ (3 / 2). Thus, the number of particles in the ground stateis

N0 ≡λ

1 − λ= ⟨N ⟩

1 −ζ (3 / 2)

ρΛ3

= ⟨N ⟩1 −

T

T0

3/2, (3.25)

where T0 is the transition temperature (i.e., where ρΛ3 = ζ (3 / 2) = 2. 61 . . .). Notice that whenT = T0 no particles are in the ground state, but this rises as temperature is lowered until 100% ofthem are in the ground state at absolute zero. Having macroscopic occupation of a single statedrastically changes the physical properties of the system. Have a look at the demonstrations byAlfred Leitner https://www.youtube.com/watch?v=sKOlfR5OcB4 for an experimental survey ofsome of the more striking changes that arise in superfluids or athttps://www.youtube.com/watch?v=BFdq6IecUJc for superconductors.

3.3.3. Photons and Black Body Radiation

Photons can be wav e-like or particle-like. The former means that each photon has a fre-quency, ω , a wav e-vector, k, and amplitudes for the components of the photon’s electric, E, andmagnetic, H, fields. The amplitudes are arbitrary, except that the radiation must be transverse,i.e., k ⋅ E = 0, while the wav e-vector and frequency are related by the well known dispersionrelation

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ω = kc, (3.26)

where c is the speed of light in vacuum. The condition of transversality means that radiation ismade up of two independent polarizations, e.g., left and right circular polarization. From theselection rule for absorption and emission, we know that the photon behaves as a spin-1 particle,modulo transversality, since a molecule’s angular momentum changes by ∆l = ±h− on absorptionor emission in all linear spectroscopies. Hence, photons are Bosons. Finally, the energy of thephoton is just ε = h− ω .

Note that the number of photons of a given frequency isn’t conserved (unlike the atoms ina chemical reaction) and, in particular, processes like

M photons at frequency ω →← N photons at frequency ω (3.27)

are possible. At equilibrium, the free energy is a minimum, and the usual argument requires that(N − M)µPhoton = 0, or µPhoton = 0.

With this introduction, consider the radiation in a closed cubical container of side length L.The radiation in the box comes to equilibrium with the walls and forms standing wav es in differ-ent directions and having different frequencies. If we assume that the electric field vanishes atthe walls and consider the field as sin-like (just like the particle in a box) it follows thatk = π /L(nx,ny,nz)

T , where ni = 1, 2, 3, . . .. By repeating the steps used above, it follows thatn(ω ) the density of radiation between frequencies ω and ω + dω is

n(ω ) =V

π 2c3

ω 2

eβ h−ω − 1, (3.28)

where remember that the photon energy states are doubly degenerate owing to the two polariza-tion possibilities. The main differences between this and our earlier result are caused by the factthat ε = h− ω = h− kc. By multiplying this result by the photon energy, h− ω , giv es

E(ω ) = Vh−

π 2c3

ω 3

eβ h−ω − 1, (3.29)

where E(ω )dω is the energy found in the frequency interval ω to ω + dω and is known asPlanck’s formula.

Limiting behaviors are easy to find; specifically,

E(ω ) ∼

VkBTω 2

π 2c3for h− ω << kBT

Vh−

π 2c3ω 3e−β h−ω when h− ω >> kBT .

(3.30)

The low frequency (or high temperature) behavior is known as the Raleigh-Jeans radiation law,and was known before the discovery of quantum mechanics. One of the problems with the clas-sical theory, according to the Raleigh-Jeans formula, is that the total radiation energy in the boxdiverges; this is known as the ultraviolet catastrophe, and posed a very difficult problem forclassical mechanics and electrodynamics.

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By integrating the full result, we find that

Etotal

V=

h−

π 2c3 ∫∞0

dωω 3

eβ h−ω − 1=

(kBT )4

(ch−)3π 2 ∫∞0

dxx3

ex − 1=

6ζ (4)(kBT )4

(ch−)3π 2, (3.31)

where the last two results were obtained by letting ω = kBT /h− x. The T 4 dependence wasdeduced classically using thermodynamics by Stephan and Wein. Wein actually proved some-thing more general called Wein’s Law; namely, E(ω )/V = ω 3 f (ω /T ), where f is not known.Note that Wein’s Law is obeyed by the Planck formula.

3.4. Appendix A: CV in an Ideal Bose or Fermi Gas

The mathematical details leading to the constant volume heat capacity, CV , cf. Eq. (3.8), are nowpresented. First, from thermodynamics, recall that

CV ≡

∂⟨E⟩∂T

⟨N ⟩,V

=

∂⟨E⟩∂T

β µ,V

+

∂⟨E⟩∂β µ

β ,V

∂β µ

∂T

⟨N ⟩,V

=

∂⟨E⟩∂T

β µ,V

−

∂⟨E⟩∂β µ

β ,V

∂⟨N ⟩∂T

β µ,V

∂⟨N ⟩∂β µ

β ,V

, (A1)

where the second equality follows when we consider ⟨E⟩ to be a function of T or β , β µ, and vol-ume, V , while the last equality is obtained when we use the cyclic rule (implicit function differ-entiation).

Next we use Eqs. (3.5)−(3.7) to evaluate the derivatives that appear in Eq. (A1); i.e.,

∂⟨E⟩∂T

β µ,V

=1

kBT 2iΣ ⟨ni⟩

2eβ (ε i−µ)ε 2

i ,

∂⟨E⟩∂β µ

β ,V

=iΣ ⟨ni⟩

2eβ (ε i−µ)ε i ,

∂⟨N ⟩∂T

β µ,V

=1

kBT 2iΣ ⟨ni⟩

2eβ (ε i−µ)ε i ,

∂⟨N ⟩∂β µ

β ,V

=iΣ ⟨ni⟩

2eβ (ε i−µ),

which when used in Eq. (A1) gives

CV =1

kBT 2iΣ ⟨ni⟩

2eβ (ε i−µ)

ε 2i − ε i

jΣ ⟨n j⟩

2eβ (ε j−µ)ε j

jΣ ⟨n j⟩

2eβ (ε j−µ)

= i, jΣ ⟨ni⟩

2eβ (ε i−µ)⟨n j⟩

2eβ (ε j−µ)ε i(ε i − ε j)

kBT 2

jΣ ⟨n j⟩

2eβ (ε j−µ)

.

(A2)

Finally, we take half the double sum in the numerator of Eq. (A2), exchange the dummy sumindices, i→← j and add it to the unmodified half, resulting in Eq. (3.8). Note that there are

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alternate ways of writing Eq. (A2). For example,

⟨ni⟩2eβ (ε i−µ) =

1

4

sech((β (ε i − µ) / 2)) for Fermions

csch((β (ε i − µ) / 2)) for Bosons.(A3)

An alternate route to CV , is to eliminate µ in favor of ⟨N ⟩ in the energy. From Eq. (3.18)we see that

ρ = D ∫∞0

dεε 1/2

eβ (ε −µ) + 1, (A4)

where ρ ≡ ⟨N ⟩/V and D ≡ 2π (2S + 1)(2m/h2)3/2. By integrating by parts, we can rewrite Eq.(A4) as

ρ =2D

3

ε 3/2

eβ (ε −µ) + 1)

∞

0

+2D

3β ∫

∞0

dεε 3/2eβ (ε −µ)

[(eβ (ε −µ) + 1]2, (A5a)

=2Dµ3/2

3 ∫∞−β µ

dx1 +

kBTx

µ

3/2ex

(ex + 1)2, (A5b)

where the boundary term in Eq. (A5a) vanishes and where we have changed variables tox ≡ β (ε − µ) to get Eq. (A5b). By using the Taylor expansion

(1 + x)α = 1 +∞

n=1Σ α (α − 1) . . . (α − n + 1)

n!xn, (A6)

we see that

ρ ∼ 2Dµ3/2

3

1 +

∞

n=1Σ 3

2

1

2. . .

5

2− n

kBT

µ

n

Zn

, (A7)

where

Zn ≡1

n! ∫∞−∞

dxxnex

(ex + 1)2=

0 for n odd

2(1 − 2−n)ζ (n), for n even,

(A8)

where ζ (n) is the Riemann zeta function.9 Note that the odd n terms vanish because ex /(ex + 1)2

is even in x.

In obtaining Eq. (A7) we’ve extended the lower integration limit to −∞. On one hand, this

only introduces exponentially small errors, since ∫−β µ

−∞dx xne−x . . . is exponentially small when

β µ >> 1; Nonetheless, while this makes sense for each term in the Taylor expansion, the sum

9See I.S. Gradshetyn and I.M. Ryzhik, op. cit., Eq. (3.411.3), p. 325.

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must (and does) diverge since (1 + kBTx/µ)3/2 is imaginary when x < −β µ. Our approach gener-ates what is known as an asymptotic expansion. If you plot the individual terms, they willdecrease until some minimum value is obtained, and increase thereafter causing the series todiverge. The trick is to sum no further. Another example of this is the Stirling formula for n!.

The goal of this calculation is to obtain a low temperature expansion for µ; accordingly, welet

µ ≡ µ0(1 + c1τ + c2τ 2 + . . .), (A9)

where µ0 ≡ h2(6ρ /π (2S + 1))2/3/8m is the chemical potential at absolute zero (cf. Eq. (A19)) andwhere τ ≡ π kBT /µ0 is a reduced temperature. When Eq. (A9) is substituted into Eq. (A7),theresult expanded into a series in τ , and the coefficients of τ n, n > 0, set to zero, it follows that10

µ = µ01 −

1

12τ 2 −

1

80τ 4 −

247

25920τ 6 −

16291

777600τ 8 −

1487

15360τ 10 + . . .

. (A10)

By starting with the general expression for the energy, i.e.,

⟨E⟩ = DV ∫∞0

dεε 3/2

eβ (ε −µ) + 1,

and repeating the steps that led from Eq. (A4) to (A7), it follows that

⟨E⟩ =2DV µ5/2

5

1 +

∞

n=1Σ 5

2

3

2. . .

7

2− n

kBT

µ

n

Zn

. (A11)

Finally, by using Eq. (A10) to eliminate µ it follows that

⟨E⟩ =3

5⟨N ⟩µ0

1 +

5

12τ 2 −

1

16τ 4 −

1235

36288τ 6 −

10367

155520τ 8 −

1478

5120τ 10 + . . .

, (A12)

which implies that

CV

⟨N ⟩kB

=π τ2

1 −

3

10τ 2 −

247

1008τ 4 −

10367

16200τ 6 −

4461

1280τ 8 + . . .

. (A13)

Note that CV ∼T as T → 0, which is consistent with the 3rd Law of Thermodynamics. The firstcorrection is O(T 3), just like the Debye-T 3 law for vibrations. That being the case, how can wedistinguish between vibrations and electronic contributions?

10The algebra becomes horrendous if you want to go much beyond the first correction; nonetheless, it’seasy for a symbolic algebra program, e.g., Mathematica or Maxima (http://maxima.sourceforge.net), to doso. Maxima was used here.

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Fig. 3.5. The low temperature, constant volume, heat capacity for asystem of ideal Fermions in a box as a function of the reduced tempera-ture, τ , introduced in the text.. The different curves correspond to thenumber of terms kept in Eq. (A13). As expected, all the curves reduceto the linear one as τ → 0. This doesn’t happen at higher temperatures,and moreover, it is obvious that something is breaking down sinceCV > 0.

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Chemistry 593 -26- The Semi-Classical Limit

4. The Semi-Classical Limit

4.1. The Semi-Classical Limit: Quantum Corrections

Here we will work out what the leading order contribution to the canonical partition func-tion in the so-called Boltzmann Statistics approximation to account for quantum exchange sym-metries. For concreteness, and in order to not make a cumbersome notation even worse, we’llconsider an interacting set of structureless particles. Our starting point is a general, albeit formal,expression for the canonical partition function, i.e.,

Q = Tre−β H

, (4.1)

where H is the Hamiltonian and β ≡ 1/kBT . The assumption that the particles are structurelessallows us to consider a Hamiltonian of the form

H ≡ −h−2

2m∇N ⋅ ∇N + V (rN ). (4.2)

The trace is invariant under unitary transformations and so we can use any representation in eval-uating it; here we will use the momentum eigenfunctions as a basis. That is

ψpN (rN ) ≡1

h3NeipN ⋅rN /h−, (4.3)

where pN are the momentum eigenvalues. It follows that

Q =1

h3N N ! ∫ dX N e−ipN ⋅rN /h−e−β H eipN ⋅rN /h−, (4.4)

where dX N ≡ dp1. . . dpN dr1

. . . drN and the factor of N ! approximately accounts for theexchange symmetries in what is known as Boltzmann Statistics.

The calculation simplifies slightly if we consider the β Laplace transform of Q; i.e.,

Q(z) ≡ ∫∞0

d β e−β zQ(β ), (4.5)

or, by using Eq. (4.4),

h3N N !Q(z) = ∫ dX N e−ipN ⋅rN /h− 1

A − BeipN ⋅rN /h−, (4.6)

where

* Our approach follows that of J.-P. Hansen and I.R. McDonald, Theory of Simple Liquids (AcademicPress, NY, 1976) Sec. 6.10.

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The Semi-Classical Limit -27- Chemistry 593

A ≡ z + Hclassical (4.7a)

and

B ≡1

2m

h−2 ∇N ⋅ ∇N + pN ⋅ pN

. (4.7b)

The classical Hamiltonian Hclassical ≡ pN ⋅ pN /2m + V (rN ) and B is the difference between theclassical and quantum mechanical kinetic energies. Clearly B and must be small if a classicaldescription is justifiable.

The operator 1/(A − B) is known as the resolvent operator and is easily shown to obey thefollowing equality

1

A − B=

1

A+

1

AB

1

A − B(4.8a)

=1

A+

1

AB

1

A+

1

AB

1

AB

1

A+ . . . (4.8b)

where the second equality is obtained by iterating the first and is nothing more than an operatorgeneralization of the geometric series. When Eq. (4.8b) is used in Eq. (4.6) we see that the firstterm results in the Laplace transform of the classical partition function, and thus,

h3N N ![Q(z) − Qclassical(z)] = ∫ dX N e−ipN ⋅rN /h−

1

AB

1

A+

1

AB

1

AB

1

A+ . . .

eipN ⋅rN /h−. (4.9)

By using Eqs. (4.7a) and (4.7b) we see that

1

AB

1

AeipN ⋅rN /h− =

eipN ⋅rN /h−

A

1

2m

h−2 ∇N ∇N 1

A+ 2ih− pN ⋅ ∇N 1

A

(4.10)

Note that we got both 1st and 2nd (in h−) order terms. The terms O(h−0) cancel! When this resultis used in the first term in Eq. (4.9) we obtain

−h−2

2m ∫ dX N∇N 1

A

2

, (4.11)

where we have integrated by parts to move one of the ∇N ’s to the left. The purely imaginaryterms linear in pN vanish when integrated since the integrand is odd in the momenta. (Rememberthat the partition function is real and so we expect this to happen more generally). Unfortunately,we’re not quite done, since the B2 terms in Eq. (4.9) also contribute at O(h−2). By noting that B isHermitian, it follows that

∫ dX N e−ipN ⋅rN /h− 1

AB

1

AB

1

AeipN ⋅rN /h− = ∫ dX N 1

A

B1

AeipN ⋅rN

2

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=h−2

m2 ∫ dX N 1

A

pN ⋅ ∇N 1

A

2

+ O(h−3). (4.12)

By combining these terms we find that

h3N N ![Q(z) − Qclassical(z)] = −h−2

2m ∫ dX N

∇N 1

A

2

−2

mA

pN ⋅ ∇N 1

A

2

+ O(h−4). (4.13)

We can now inv ert the Laplace transforms by making use of the convolution theorem,which states that the Laplace transform of a convolution integral,

f * g ≡ ∫τ

0f (t − τ )g(τ )

is

˜f * g = f g.

Thus, returning to Eq. (4.13), we see that

h3N N ![Q − Qclassical] = −h−2

2m ∫ dX N ∫β

0d β1 e−β H

(β − β1)β1FN ⋅ FN

−2

m ∫β1

0d β2

pN ⋅ FN

2

(β − β1)(β1 − β2)

+ O(h−4),

where FN ≡ −∇NH is force acting on the particles. The various β integrals are easy to do and

we find that

h3N N ![Q − Qclassical] = −h−2 β 3

12m ∫ dX N e−β H FN ⋅ FN −

β2m

(pN ⋅ FN )2. (4.14)

Since momentum and coordinates are separable classically, and moreover the momentum distri-bution is just the Maxwell-Boltzmann distribution, we can carry out all the momentum integra-tions and find that

Q − Qclassical ∼ −

2π mkBT

h2

3N /2h−2 β 3

24mN ! ∫ drN e−βV FN ⋅ FN , (4.15)

and finally,

Q ∼ Qclassical1 −

h−2 β 3

24m⟨FN ⋅ FN ⟩

, (4.16)

where

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Qclassical =

2π mkBT

h2

3N /2ZN

N !(4.17)

with

ZN ≡ ∫ drN e−βV (4.18)

which is known as the configurational partition function.

This result is strange. If the particles are identical we can write

⟨FN ⋅ FN ⟩ =j

Σ ⟨F j ⋅ F j⟩ = N ⟨F1 ⋅ F1⟩, (4.19)

where F1 is the total force exerted on particle 1. The correction is huge, unless⟨F1 ⋅ F1⟩ ∼ O(N −1) which seems unlikely. Fortunately, the explicit factor of N is just what isneeded to make the extensive thermodynamic functions extensive. To see this, remember that theHelmholtz Free Energy, A, is related to the partition function as A = −kBT ln(Q), which fromEq. (4.16), can be written as

A = Aclassical − kBT ln1 − N

h−2 β 3

24m⟨F1 ⋅ F1⟩ + O(h−4)

(4.20)

The expansion we just carried out above assumes that h− is small parameter and has O(h−4) correc-tions that were dropped. The logarithm also has similar corrections; specifically, by noting thatthe Taylor expansion of ln(1 − x) is

ln(1 − x) = −x −x2

2. . . −

xn

n. . . ,

which when applied to Eq. (4.20) shows that the x2 term is O(h−4). Given that we’ve alreadydropped O(h−4) terms in arriving at Eq. (4.20), consistency requires that we not keep them in thelogarithm. Thus,

β (A − Aclassical)

N∼ h−2 β 3

24m⟨F1 ⋅ F1⟩ + O(h−4) ∼ Λ2 β 2

48π⟨F1 ⋅ F1⟩ + O(h−4), (4.21)

where the last expression arises when we note that h−2 = Λ2mkBT /2π , where Λ is the thermal deBroglie wav elength. Of course, what remains to be shown is that the O(h−4) terms that arise bothfrom the expansion of the trace and the logarithm have enough cancellations to give a result thatis O(N ). This is the case, as we will see later in a more general context.

We can make sev eral observations about our result. First, note that the leading order freeenergy correction is positive, no matter what the interaction potential. Second, Eq. (4.21) can berewritten as


N∼ Λ2 β

48π ∫drN

ZN

F1 ⋅ ∇1e−βV (rN ) = −Λ2 β48π ∫ drN e−βV (rN )

ZN

∇1 ⋅ F1, (4.22)

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where the last equality is obtained by integrating by parts. To go further, we’ll assume that theinteractions are pairwise additive; i.e.,

Fi =j≠iΣ Fi, j(ri, j), (4.23)

where ri, j ≡ ri − r j and Fi, j is the force particle j exerts on i. With this, given that the particlesare identical, we can rewrite Eq. (4.22) as


N∼ − (N − 1)

Λ2 β48π ∫ dr1dr2

∂F1,2

∂r1,2

∫ dr3

. . . drN

e−βV (rN )

ZN

, (4.24)

where the factor of (N − 1) is just the number of ways of choosing the particle interacting withparticle 1. By introducing the so-called generic reduced pair correlation function, ρ (2)(r1, r2) andpair correlation function, g(2)(r1, r2), i.e.,

ρ (2)(r1, r2) ≡ ρ2g(2)(r12) ≡ N (N − 1) ∫ dr3. . . drN

e−βV (rN )

ZN

, (4.25)

where ρ ≡ N /V is the number density, we can rewrite Eq. (4.24) as


N∼ −

Λ2 ρ2

48π N ∫ dr1dr2

∂F1,2

∂r1,2

g(2)(r1,2)

= −Λ2 β ρ48π ∫ dr1,2

∂F1,2

∂r1,2

g(2)(r1,2), (4.26)

where the last equality is obtained by going to a coordinate system centered on r2 for the r1 inte-gration and assuming that the system is translationally invariant. The subsequent integration inr2 gives a factor of volume. Finally, by writing F1,2(r1,2) = −∇1,2u1,2(r1,2), where u1,2(r1,2) is thepair potential and only depends on the distance between particles 1 and 2, Eq.(26) can be rewrit-ten as


N∼ Λ2 β ρ

48π ∫ dr1,2 [∇21,2u1,2(r1,2)]g(2)(r1,2) (4.27a)

∼ Λ2 β ρ12 ∫

∞0

dr r2

∂2u1,2(r)

∂r2

+2

r

∂u1,2(r)

∂r

g(2)(r),(4.27b)

where ∇21,2 is the Laplacian for r1,2. The 2nd equality is obtained by changing the integration and

Laplacian to polar coordinates and letting r1,2 → r.

The energies that matter are those O(kBT ); hence, a crude estimate of size of our correc-tion is Λ2 ρσ = ρΛ3σ /Λ , where σ is the molecular interaction length scale. The use of Boltz-mann Statistics already requires that ρΛ3 << 1, and thus, unless σ /Λ >> 1, the semi-classical

Winter, 2018


approximation should be valid. The proceeding estimates are crude. Hansen and Weis* per-formed monte carlo simulations for a model of neon near the triple point and showed that thecorrections are about 1% of the Boltzmann statistic’s classical result.

*J.P. Hansen and J.J Weis, Phys. Rev. 188, 314 (1969).

Winter, 2018

Chemistry 593 -32- Dense Gases: Virial Coefficients

5. Dense Gases: Virial Coefficients

One way to correct the ideal gas equation of states is to use the virial expansion, namely,

β p = ρ + B2(T )ρ2 + B3(T )ρ3+. . . , (5.1)

where ρ ≡ N /V is the number density and where Bn(T ) is known as the n’th virial coefficient.Our goal is to obtain a molecular expression for the virial coefficients (at least for B2(T )).

To proceed, we need to identify a useful small parameter that can be used to expand theequation of state. To do this, note that in Boltzmann statistics and ignoring all interactionsbetween molecules, the canonical partition function for a system of N molecules,QN (V , T ) = [q1(T )V ]N /N !, where V q1(T ) is the molecular partition function. Hence, the grandpartition function becomes

Ξ =∞

N=0Σ eβ µN [V q1(T )]N

N != exp

λV q1(T )

, (5.2)

where λ ≡ eβ µ, is the activity. By taking the appropriate derivatives of log(Ξ), it follows that

ρ ≡⟨N ⟩V

=1

V

∂ log(Ξ)

∂β µ

β ,V

=λV

∂ log(Ξ)

∂λβ ,V

= λ q1(T ), (5.3)

where the last equality is obtained when Eq. (5.2) is used. The ideal gas equation of state arisessince β p = V −1 log(Ξ) = λ q(T ) = ρ . Thus, we see that λ∝ρ as ρ → 0, where ideal gas behavioris expected. Since, in general,

Ξ = 1 +∞

N=1Σ λ N QN (V , T ), (5.4)

we can view the grand partition function as a power series in the small parameter λ , at least forlow densities.

Unfortunately, λ is not a particularly convenient quantity to control experimentally (unlikethe density). To deal with this, assume that the activity can be written as a power series in den-sity; i.e.,

λ = ρc1 + ρ2c2 + ρ3c3+. . . . (5.5)

From Eqs. (5.3) and (5.4) it follows that

ρV =

∞

N=1Σ N λ N QN (T , V )

1 +∞

N=1Σ λ N QN (T , V )

, (5.6)

or equivalently,

Winter, 2018

Dense Gases: Virial Coefficients -33- Chemistry 593

ρV1 +

∞

N=1Σ λ N QN (T , V )

=∞

N=1Σ N λ N QN (T , V ). (5.7)

Equation (5.5) is used to rewrite Eq. (5.7) as

ρV +∞

N=1Σ ρ N+1(c1 + ρc2 + ρ2c3 + . . .)NVQN (T , V ) =

∞

N=1Σ N ρ N (c1 + ρc2 + ρ2c3 + . . .)N QN (T , V ),

(5.8)

and similar to what is done in the Frobenius method for ODE’s, we equate the coefficients ofeach power of ρ; i.e.,

V = c1Q1 for O(ρ), (5.9a)

c1VQ1 = c2Q1 + 2c21Q2 for O(ρ2), (5.9b)

c2VQ1 + c21VQ2 = c3Q1 + 4c1c2Q2 + 3c3

1Q3 for O(ρ3), (5.9c)

etc.. These equations are easily solved, giving

c1 =V

Q1

, (5.10a)

c2 = −(2Q2 − Q2

1)V 2

Q31

, (5.10b)

c3 = −(3Q1Q3 − 8Q2

2 + 5Q21Q2 − Q4

1)V 3

Q51

, (5.10c)

etc.. By substituting Eqs. (5.10a-c) into Eq. (5.5) we see that

λ =ρV

Q1

1 −

(2Q2 − Q21)ρV

Q21

−(3Q1Q3 − 8Q2

2 + 5Q21Q2 − Q4

1)ρ2V 2

Q41

+ . . ., (5.11)

which gives

β µ = log(λ) = log

ρV

Q1

−

2Q2 − Q2

1V ρ

Q21

+

Q4

1 − 6Q2Q21 − 6Q3Q1 + 12Q2

2V 2 ρ2

2Q41

+ . . . , (5.12)

expressing the non-ideal corrections to the Gibbs free energy (recall that µ = G in a one compo-nent system) in terms of canonical partition functions for systems of N = 2, 3, . . . particles; inaddition, we’ve made no assumptions about using classical or quantum mechanics. Similarly,

Winter, 2018


remembering that β pV = log Ξ, we see that

β p = ρ −(2Q2 − Q2

1)V ρ2

2Q21

−(6Q1Q3 − 12Q2

2 + 6Q21Q2 − Q4

1)V 2 ρ3

3Q41

+ . . . . (5.13)

Note that in obtaining Eqs. (5.12) and (5.13) we have had to re-expand the logs in order to beconsistent with the omitted terms.*

Our results can be simplified if we assume that the internal electronic, vibrational, androtational degrees of freedom are separable, and that the translational ones are classical. Thus

QN ≡qN

1 ZN

N !,

where, as above, q1 is the molecular partition function for all intra-molecular degrees of freedom,including translation, divided by volume, and

ZN ≡ ∫ dr1. . . ∫ drN e−βU (N )(r1,...,rN ), (5.14)

where U (N )(r1, . . . , rN ) is the interaction potential for a system of N molecules and Z1 = V . ZN

is known as the configurational partition function. Note that the separability assumption is rea-sonable for electronic (in the absence of chemical reactions) and for vibrational degrees of free-dom, but is problematic for rotations if the molecule is aspherical. Fortunately, the rotationaldegrees of freedom can be treated classically for most of the cases of interest, and can be incor-porated into the definition of ZN , leaving only the rotational kinetic energy contributions in q1

(this is analogous to what we do with the translational contributions in q1 or with the rotationalpartition function when T >> Θrot).

With this change of notation, it follows that

β µ = log

ρq1

−∞

n=1Σ β n ρ n, (5.15)

where

β1 ≡Z2 − V 2

V, (5.16a)

*Specifically, we use the Taylor expansion

log(1 + x) = x −1

2x2 +

1

3x3 + . . . =

∞

n=1Σ (−1)n+1

nxn,

where x ≡ λ /(ρV /Q1), cf. Eq. (11). Thus, e.g., if we only keep terms to O(ρ) in λ we should only keepterms through the same order in the series for the logarithm; i.e.,

β µ = log(ρV /Q1) − (2Q2 − Q21)V ρ /Q2

1 + O(ρ2).

Winter, 2018


β2 ≡Z3V + 3Z2V 2 − V 4 − 3Z2

2

2V 2, (5.16b)

etc.. The β n’s are referred to as n’th order irreducible cluster integrals. By expressing the densityexpansion of the pressure in terms of the configurational partition functions and the irreduciblecluster integrals, we see that

β p = ρ −β1

2ρ2 −

2β2

3ρ3+. . . . (5.17)

Hence, B2(T ) = −β1/2, B3(T ) = −2β2/3.

A general connection between the irreducible cluster integrals and the virial coefficientscan easily be obtained from the Gibbs-Duhem relation, SdT − Vdp + Ndµ = 0, specifically,

∂β p

∂ρ

T

= ρ

∂β µ

∂ρ

T

= 1 −∞

n=1Σ nβ n ρ n, (18)

where the second equality is obtained when (5.15) is used. By integrating, remembering thatp = 0 when ρ = 0, we see that

β p = ∫ρ

0d ρ

∂ p

∂ρ

T

= ρ −∞

n=1Σ n

n + 1β n ρ n+1, (19)

and thus,

Bn+1 = −n

n + 1β n. (5.20)

It turns out that even though our definitions of β n explicitly contain the volume V, the vol-ume dependence disappears. For example, consider the second virial coefficient, B2(T ).According to Eqs. (5.16a) and (5.20),

B2(T ) = −Z2 − V 2

2V=

1

2V ∫ dr1 ∫ dr21 − e−β u12(r12)

, (5.21)

where u12(r12) is the interaction potential of a pair of molecules separated by a distance r12. Forneutral molecules, the interaction rapidly decays to zero once r12 is much larger than a few tensof angstroms, as will the integrand in Eq. (5.21); hence, it is useful to change one of the integra-tion variables, say r1 to a basis centered on r2. This allows Eq. (5.21) to be rewritten as

B2(T ) =1

2V ∫ dr2 ∫ dr121 − e−β u12(r12)

=

1

2 ∫ dr121 − e−β u12(r12)

. (5.22a)

Similar manipulations can be used in Eq. (5.16b) to show that B3 only depends on temperature.More generally, there are some sophisticated mathematical tools that prove this in general, butthey are well beyond the level of this class.*

*See, e.g., J.P. Hansen and I.R. MacDonald, Theory of Simple Liquids, (Academic Press, London, 1976),ch. 4.

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For spherically symmetric interactions we can switch to polar coordinates, giving

B2(T ) = 2π ∫∞0

dr12 r212

1 − e−β u12(r12)

. (5.22b)

If the pair interactions are predominantly repulsive, i.e., u12(r12) > 0, the quantity in the paren-thesis will be positive, as will B2(T ) and the deviation from ideal gas behavior will be positive.If the interactions are predominantly attractive, the reverse will happen. Consider a more realis-tic potential where steric repulsions dominate at short distances and attractions dominate a largerdistances, e.g., as for the so called square well potential

u12(r) =

∞ for r < σ−ε for σ < r < γ σ0 otherwise,

where ε ≥ 0 and γ ≥ 1. The potential is shown in Fig. 5.1.

Fig. 5.1. The square well (solid) and more realistic Lennard-Jones (L-J) 6-12 (dashed) potentials, respectively. The L-J potential,u(r) = 4ε [(σ /r)12 − (σ /r)6], is more realistic, in that it includes theeffects of London dispersion forces at larger distances.

The integral in Eq. (5.22b) is easily done and gives

B2(T ) =2π σ 3

3

1 − (γ 3 − 1)(eβ ε − 1)

. (5.23)

From this it is easy to show that the Boyle temperature, where B2(TB) = 0, is

TB = −ε

kB log(1 − γ −3).

The van der Waals model gives B2(T ) = b − β a, where a and b are the van der Waals a and bcoefficients. Both models show that the effects of steric repulsion and attraction are additive.

Winter, 2018


Other than that, the temperature dependence of the attractive terms differ slightly, although theyapproach each other when β ε → 0 where eβ ε − 1∼β ε .

The integration in Eq. (5.22b) for the more realistic Lennard-Jones 6-12 potential is morecomplicated and is given in the Appendix.

Fig. 5.2. The reduced second virial coefficient for the Lennard-Jones poten-tial. B*

2 ≡ B2/B2, hard sphere, where B2, hard sphere ≡ 2π σ 3/3, is the second virialcoefficient for the hard-sphere potential, and where the reduced temperatureis T * ≡ kBT /ε . As expected B*

2 > (<) 0 for high (low) reduced temperatures,where repulsions (attractions) dominate. The reduced Boyle temperature isapproximately 3.43. Note that B*

2∝(T *)−1/4 as T * → ∞ and is not constant(why?).

The integral in our expression for B2(T ), cf. Eq. (5.22b), imposes some important limita-tions on the interaction potential, u12(r); specifically, for large separations u12(r) → 0, and thusr21 − exp[−β u12(r)]∼r2 β u12(r) showing that u12(r) must decay faster than 1/r3 in order thatthe integral converges. Charge-charge, charge-dipole, and dipole-dipole interactions decay liker−1, r−2, and r−3, respectively! Charge-dipole and dipole-dipole interactions depend on the orien-tation of the dipole moments (i.e., angular degrees of freedom), and when these are includedproperly the integral converges. Ionic interactions are more problematic; in short, the systemmust be electrically neutral, and even then, the basic assumption about the nature of the virialexpansion (a power series in density) breaks down, giving rise to ρ1/2 terms.

5.1. Appendix: B2(T ) for the Lennard-Jones potential

This section is based on the extensive discussion of the viral expansion by Hirshfelder,Curtiss, and Bird* First, express B2 in a dimensionless form by writing

*J.O. Hirshfelder, C.F. Curtiss, and R.P. Bird, Molecular Theory of Gases and Liquids, 2nd ed. (John Wileyand Sons Inc, N.Y., 1964), ch. 3.

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B2(T ) =2π σ 3

3B*

2(T *), (5.24)

where we have let r → σ r, T * ≡ kBT /ε is the reduced temperature and where the reduced 2ndvirial coefficient is

B*2(T *) ≡ 3 ∫

∞0

dr r21 − exp

−

4

T *(r−12 − r−6)

, (5.25a)

=24

T * ∫∞0

dr r2(2r−12 − r−6) exp−

4

T *(r−12 − r−6)

, (5.25b)

=24

T * ∫∞0

dr r2(2r−12 − r−6) exp−

4

T *r−12

∞

n=0Σ 1

n!

4r−6

T *

n

,(5.25c)

where the second equality is obtained by integration by parts and where the last expression isobtained by expanding exp(4r−6/T *) in a Taylor series (i.e., we treat the attractions in the Boltz-mann factor as a perturbation).

Next we change variables by letting

z ≡4

T *r−12 or r =

T *

4z

−1/12

, (5.26)

thereby implying that

dr = −

T *

4

−1/12z−13 / 12

12dz. (5.27)

When these are used in Eq. (5.25c), the latter becomes

B*2(T *) =

∞

n=0Σ 2n+1/2(T *)−(2n+3) / 4

n! ∫∞0

dz z(2n−3) / 4(√z√ T * − 1)e−z (5.28a)

=∞

n=0Σ 2n+1/2(T *)−(2n+3) / 4

n!

√ T *Γ

2n + 3

4

− Γ

2n + 1

4

(5.28b)

= −∞

n=0Σ

2n−3/2Γ

2n − 1

4

n!

1

T *

(2n+1) / 4

, (5.28c)

where

Winter, 2018


Γ(z) = ∫∞0

dt t z−1e−t , (5.29)

is known as the Γ-function. The last equality was obtained by noting that Γ(x + 1) = xΓ(x) andredefining the sum index.

Winter, 2018

Chemistry 593 -40- Correlation Functions and the Pressure

6. Correlation Functions and the Pressure

In order to calculate the pressure for liquids it is tempting to imagine that we could simplyresum the virial expansion in some way; unfortunately, this doesn’t work and an alternateapproach is required. We’ll show two methods here.

6.1. The Virial Form of The Pressure

In the canonical ensemble, the pressure can be determined from the partition as

β p =

∂ ln(Q)

∂V

N ,β=

∂ ln(ZN )

∂V

N ,β, (6.1)

where we are using classical statistical mechanics, with Q = ZN /(N !Λ3N ), ZN is the configura-tional partition function, and Λ = h/(2π mkBT )1/2 is the thermal de Broglie wav elength. The vol-ume only appears in the integration limits, and when differentiated, pins the position of one ofthe particles to one of the faces of the cubical box. Although the result is correct, it is not whatwe want, namely, an expression where the walls of the container do not play an explicit role.

An alternative expression can be found by first changing the integration variables to explic-itly scale the particle coordinates by the size of the box, thereby removing the volume from thelimits of integration, i.e., by letting rN → V 1/3rN ,*. With this, we see that

ZN = V N ∫1

0dx*

1. . . ∫

1

0dz*

N e−βU(rN ,*V 1/3). (6.2)

When this is used in Eq. (6.1) we find that

β p = ρ +βV N−1

3 ∫1

0dx*

1. . . ∫

1

0dz*

N FN ⋅ rN e−βU(rN )

ZN

(6.3a)

= ρ +β

3V⟨FN ⋅ rN ⟩ = ρ +

β N

3V⟨F1 ⋅ r1⟩ (6.3b)

= ρ +β ρ2

3V ∫ dr1dr2 F1,2 ⋅ r1 g(2)(r1, r2), (6.3c)

where ρ = N /V , and where the last equality assumes a pairwise additive potential. Newton’saction-reaction law demands that F1,2 = −F2.1, thereby allowing us to switch the integration vari-ables in Eq. (6.3c) in half the integral, which becomes

β p = ρ +β ρ2

6 ∫ dr1,2 F1,2 ⋅ r1,2 g(2)(r1,2). (6.4)

This result is known as the virial form of the pressure (not to be confused with the virialexpansion).

Winter, 2018

Correlation Functions and the Pressure -41- Chemistry 593

6.2. The Compressibility Form of the Pressure

There is an alternate route to the pressure based on the reduced generic distribution func-tions in the grand canonical ensemble. First, note that

ρ (m)GC (r1, . . . , rm) ≡

NΣ P(N )ρ (m)

C (r1, . . . , rm), (6.5)

where P(N ) = eβ µN Q(N , V , T )/Ξ is the probability of having a system with N particles in the

grand canonical ensemble, and ρ (m)C is the generic reduced distribution function in the canonical

ensemble with N particles. In general,

∫ dr1. . . drm ρ (m)

C (r1, . . . , rm) =N !

(N − m)!(6.6a)

and

∫ dr1. . . drm ρ (m)

GC (r1, . . . , rm) = < N !

(N − m)! >GC. (6.6b)

When m = 2, remembering that ρ (2)GC (r1, r2) ≡ ρ2g

(2)GC (r1, r2), we rewrite Eq. (6.6b) as

ρ2 ∫ dr1dr2 g(2)GC (r1, r2) = < N (N − 1) >GC= < N 2 >GC − < N >GC . (6.7)

By adding and subtracting 1 to the integrand and rearranging the result, Eq. (6.7) becomes

< N > +ρ2 ∫ dr1dr2g(2)(r1, r2) − 1

= < N 2 > − < N >2 , (6.8)

where the GC labels have been dropped; henceforth, all averages and correlation functions aregrand canonical. The right hand side of the equation is the variance of the number distribution inthe grand canonical ensemble; i.e.,

< (N− < N >)2 > = < N 2 > − < N >2 = ρkBTκ < N > ,

where κ ≡ −V −1(∂V /∂P)N ,T is the isothermal compressibility. Thus, Eq. (6.8) becomes

1 + ρ ∫ dr1,2g(2)(r1,2) − 1

= ρkBTκ , (6.9)

By noting that κ = ρ−1(∂ρ /∂P)T , we can rewrite Eq. (6.9) as

1 + ρ ∫ dr1,2g(2)(r1,2) − 1

=

∂ρ∂β P

β

, (6.10a)

or equivalently,

Winter, 2018

Chemistry 593 -42- Correlation Functions and the Pressure

∂β P

∂ρ

T

=1

1 + ρ ∫ dr1,2g(2)(r1,2) − 1

. (6.10b)

This is known as the compressibility form of the equation of state (it is easily integrated overdensity to get the pressure). Note that we have not made any assumptions about the form of theinteraction potential, e.g., pairwise additivity. As we shall see later, the integral involving thepair correlation function can be measured in a variety of scattering experiments, which whenused in Eq. (6.10b), yield the compressibility (and hence the pressure).

In the end, the two approaches considered here should yield equivalent results, assuming,of course, that the pair correlation function is correct. This is never the case owing to inaccura-cies in the theories used to calculate the pair correlation function. Nonetheless, approximate the-ories of g(2)(r) can be judged on how many virial coefficients of either form are correct. Alter-nately, sev eral "generalized" approaches add an adjustable parameter that is used to force equiva-lency of the two expressions. It is still not clear how or why this leads to a more accurate pres-sure.

Winter, 2018

Normal Mode Analysis -43- Chemistry 593

7. Normal Mode Analysis

7.1. Quantum Mechanical Treatment

Our starting point is the Schrodinger wav e equation:

−

N

i=1Σ h−2

2mi

∂2

∂→r

2i

+ U(→r1, . . . ,

→r N )

Ψ(

→r1, . . . ,

→r N ) = E Ψ(

→r1, . . . ,

→r N ), (7.1.1)

where N is the number of atoms in the molecule, mi is the mass of the i’th atom, andU(

→r1, . . . ,

→r N ) is the effective potential for the nuclear motion, e.g., as is obtained in the Born-

Oppenheimer approximation.

If the amplitude of the vibrational motion is small, then the vibrational part of the Hamil-tonian associated with Eq. (7.1.1) can be written as:

Hvib ≈ −N

i=1Σ h−2

2mi

∂2

∂→∆

2

i

+ U0 +1

2

N

i, j=1Σ

↔Ki, j :

→∆i

→∆ j , (7.1.2)

where U0 is the minimum value of the potential energy,→∆i ≡ →

r i −→Ri ,

→Ri is the equilibrium posi-

tion of the i’th atom, and

↔Ki, j ≡

∂2U

∂→r i∂

→r j

→

r k =→Rk

(7.1.3)

is the matrix of (harmonic) force constants. Henceforth, we will shift the zero of energy so as tomake U0 = 0. Note that in obtaining Eq. (7.1.2), we have neglected anharmonic (i.e., cubic andhigher order) corrections to the vibrational motion.

The next and most confusing step is to change to matrix notation. We introduce a columnvector containing the displacements as:

∆ ≡ [∆x1 , ∆y

1 , ∆z1, . . . , ∆x

N , ∆yN , ∆z

N ]T , (7.1.4)

where "T " denotes a matrix transpose. Similarly, we can encode the force constants or massesinto 3N × 3N matrices, and thereby rewrite Eq. (7.1.2) as

Hvib = −h−2

2

∂∂∆

†

M−1

∂∂∆

+1

2∆†K∆, (7.1.5)

where all matrix quantities are emboldened, the superscript "†" denotes the Hermitian conjugate(matrix transpose for real matrices), and where the mass matrix, M, is a diagonal matrix with themasses of the given atoms each appearing three times on the diagonal.

The Hamiltonian given by Eq. (7.1.5) is the generalization of the usual harmonic oscillatorHamiltonian to include more particles and to allow for "springs" between arbitrary particles.

Winter, 2018

Chemistry 593 -44- Normal Mode Analysis

Unless K is diagonal (and it usually isn’t) Eq. (7.1.5) would seem to suggest that the vibrationalproblem for a polyatomic is non-separable (why?); nonetheless, as we now show, it can be sepa-rated. This is first shown using using the quantum mechanical framework we’ve just set up.Later, we will discuss the separation using classical mechanics. This is valid since, for harmonicoscillators, you get the same result for the vibration frequencies.

We now make the transformation→∆i → m−1/2

i

→∆i , which allows Eq. (7.1.5) to be reexpressed

in the new coordinates as

Hvib = −h−2

2

∂∂∆

†

∂∂∆

+1

2∆†K∆, (7.1.6)

where

K ≡ M−1/2KM−1/2. (7.1.7)

Since the matrix K is Hermitian (or symmetric for real matrices), it is possible to find aunitary (also referred to as an orthogonal matrix for real matrices) matrix which diagonalizes it;i.e., you can find a matrix P which satisfies

P†KP = λ or K = PλP†, (7.1.8)

where λ is diagonal (with real eigenvalues) and where

P P† = 1 (7.1.9)

where 1 is the identity matrix. Note that P is the unitary matrix who’s columns are the normal-ized eigenvectors. (See a good quantum mechanics book or any linear algebra text for proofs ofthese results).

We now make the transformation

∆ → P∆ (7.1.10)

in Eq. (7.1.5). This gives

Hvib = −h−2

2

∂∂∆

†

P†P

∂∂∆

+1

2∆† λ∆, (7.1.11)

Finally, Eq. (7.1.8) allows us to cancel the factors of P in this last equation and write

Hvib =3N

i=1Σ

−

1

2h−2 ∂2

∂∆2i

+1

2λ i∆2

i. (7.1.12)

Thus the two transformations described above hav e separated the Hamiltonian into 3N uncoupledharmonic oscillator Hamiltonians, and are usually referred to as a normal mode transformation.Remember that in general the normal mode coordinates are not the original displacements fromthe equilibrium positions, but correspond to collective vibrations of the molecule.

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7.2. Normal Modes in Classical Mechanics

The starting point for the normal mode analysis in classical mechanics are Newton’s for asystem of coupled harmonic oscillators. Still, using the notation that led to Eq. (7.1.12), we canwrite the classical equations of motion as:

M∆(t) = −K∆(t), (7.2.1)

where components of the left-hand-side of the equation are the rates of change of momentum ofthe nuclei, while the right-hand-side contains the harmonic forces.

Since we expect harmonic motion, we’ll look for a solution of the form:

∆(t) = cos(ω t + φ )Y, (7.2.2)

where φ is an arbitrary phase shift. If Eq. (7.2.2) is used in Eq. (7.2.1) it follows that:

(Mω 2 − K)Y = 0, (7.2.3)

where remember that Y is a column vector and M, and K are matrices. We want to solve thishomogeneous system of linear equations for Y. In general, the only way to get a nonzero solu-tion is to make the matrix [Mω 2 − K] singular; i.e., we must set

det(Mω 2 − K) = det(ω 2 − M−1/2KM−1/2) det(M) = 0. (7.2.4)

Since, det(M) = (M1 M2. . . MN )3 ≠ 0 we see that the second equality is simply the characteristicequation associated with the eigenvalue problem:

Ku = λu, (7.2.5)

with λ ≡ ω 2 and with K given by Eq. (7.2.7) above. The remaining steps are equivalent to whatwas done quantum mechanically.

7.3. Force Constant Calculations

Here is an example of a force constant matrix calculation. We will consider a diatomicmolecule, where the two atoms interact with a potential of the form:

U(r1, r2) ≡1

2K

|r1 − r2| − R0

2

; (7.3.1)

i.e., a simple Hookian spring. It is easy to take the various derivatives indicated in the precedingsections; here, however, we will explicitly expand the potential in terms of the atomic displace-ments, ∆i . By writing, ri = Ri + ∆i (where Ri is the equilibrium position of the i’th nucleus), Eq.,(7.3.1) can be rewritten as:

U(r1, r2) =1

2K

R2

12 + 2R12 ⋅ (∆1 − ∆2) + |∆1 − ∆2|2

1/2

− R0

2

, (7.3.2)

Winter, 2018


where R12 ≡ R1 − R2. Clearly, the equilibrium will have |R12| = R0. Moreover, we expect thatthe vibrational amplitude will be small, and thus, the terms in the ∆’s in the square root in Eq.(7.3.2) will be small compared with the first term. The Taylor expansion of the square rootimplies that

√ A + B ≈ √ A +B

2√ A+. . . , (7.3.3)

we can write

U(r1, r2) =1

2K

R12 +

2R12 ⋅ (∆1 − ∆2) + |∆1 − ∆2|2

2R12

− R0

2

, (7.3.4)

which can be rewritten as

U(r1, r2) =1

2K [R12 ⋅ (∆1 − ∆2)]2, (7.3.5)

where the ˆ denotes a unit vector and where all terms smaller than quadratic in the nuclear dis-placements have been dropped. If the square is expanded, notice the appearance of cross terms

in the displacements of 1 and 2.

It is actually quite simple to finish the normal mode calculation in this case. To do so,define the equilibrium bond to point in along the x axis. Equation (7.3.5) shows that only x-components of the displacements cost energy, and hence, there will no force in thy y or z direc-tions (thereby resulting in 4 zero eigen-frequencies). For the x components, Newton’s equationsbecome:

m1

0

0

m2

⋅¨

∆x

1

∆x2

= −

K

−K

−K

K

⋅ ∆x

1

∆x2

, (7.3.6)

where mi is the mass of the i’th nucleus. This in turn leads to the following characteristic equa-tion for the remaining frequencies:

0 = det

m1

0

0

m2

ω 2 −

K

−K

−K

K

= (m1ω 2 − K )(m2ω 2 − K ) − K 2

and

= ω 2(µω 2 − K ),

where µ ≡ m1m2/(m1 + m2) is the reduced mass. Thus we pick up another zero frequency and a

nonzero one with ω = √ K /µ, which is the usual result. (Note that we don’t count ± rootstwice--WHY?).

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7.4. Normal Modes in Crystals

The main result of the preceding sections is that the characteristic vibrational frequenciesare obtained by solving for eigenvalues, cf., Eq. (7.2.5). For small to mid-size molecules this canbe done numerically, on the other hand, this is not practical for crystalline solids where the matri-ces are huge (3N × 3N with N∼O(N A))!

Crystalline materials differ from small molecules in one important aspect; they are peri-odic structures made up of identical unit cells; specifically, each unit cell is placed at

→R = n1

→a1 + n2

→a2 + n3

→a3, (7.4.1)

where ni is an integer and→ai is a primitive lattice vector. The atoms are positioned within each

unit cell at positions labeled by an index, α . Thus, the position of any atom in entire crystal isdetermined by specifying R and α . Hence, Eq. (7.2.5) can be rewritten as

λuR,α =R′Σ

α ′Σ KR,α ;R′,α ′uR′,α ′. (7.4.2)

The periodicity of the lattice implies that only the distance between unit cells can matter; i.e.,KR,α ;R′,α ′ = KR−R′;α ,α ′, and this turns Eq. (7.4.2) into a discrete convolution that can be simplifiedby introducing a discrete Fourier transform, i.e., we let

uk,α ≡RΣ eik⋅RuR,α . (7.4.3)

When applied to both sides of Eq. (7.2.5), this gives*

λ uk,α ≡αΣ Kk;α ,α ′uk,α ′, (7.4.4)

where

Kk;α ,α ′ ≡RΣ eik⋅RKR;α ,α ′. (7.4.5)

The resulting eigenvalue problem has rank 3Ncell , where Ncell is the number of atoms in a unitcell, and is easily solved numerically.

We hav en’t specified the values for the k’s. It turns out that a very convenient choice is to

use→k’s expressed in terms of the so-called reciprocal lattice vectors,

→G, i.e.,

→G ≡ m1

→b1 + m2

→b2 + m3

→b3, (7.4.6)

where the mi’s are integers, and where the reciprocal lattice basis vectors are defined as†

*Strictly speaking, in obtaining Eq. (7.4.5) we have assumed some special properties of the lattice. Wedon’t use a finite lattice, rather one that obeys periodic boundary conditions (see below).†Here we’re using the definition of C. Kittel, Introduction to Solid State Physics (Wiley, 1966), p. 53.There are others, see, e.g., M. Born and K. Huang, Dyamical Theory of Crystal Lattices (Oxford, 1968), p.69.

Winter, 2018


→b1 ≡ 2π

→a2 × →

a3

|→a1 ⋅ (

→a2 × →

a3)|,

→b2 ≡ 2π

→a3 × →

a1

|→a2 ⋅ (

→a3 × →

a1)|, and

→b3 ≡ 2π

→a1 × →

a2

|→a3 ⋅ (

→a1 × →

a2)|. (7.4.7)

Note the following:

a) The denominators appearing in the definitions of the→bi are equal; i.e.,

|→a1 ⋅ (

→a2 × →

a3)| = |→a2 ⋅ (

→a3 × →

a1)| = |→a3 ⋅ (

→a1 × →

a2)| = ν cell , (7.4.8)

where ν cell is the volume of the unit cell. The volume of the reciprocal lattice unit cell is(2π )3/ν cell .

b) By construction,

→ai ⋅

→b j = 2π δ i, j , (7.4.9)

where δ i, j is a Kronecker-δ .

c) The real and reciprocal lattices need not be the same, even ignoring how the basis vectorsare normalized; e.g., they are for the SCC, but the reciprocal lattice for the BCC lattice is aFCC lattice.

d) For any lattice vector,→R, and reciprocal lattice vector,

→G,

ei→R⋅

→G = e2π i(m1n1+m2n2+n3m3) = 1, (7.4.10)

cf. Eqs. (7.4.1), (7.4.6), and (7.4.9). Hence, adding any reciprocal a lattice vector to the→k

in Eq. (7.4.4) changes nothing, and so, we restrict→k to what is known as the First Brillouin

Zone; specifically,

→k = k1

→b1 + k2

→b2 + k3

→b3, (7.4.11)

where ki ≡ mi /Ni with mi = 0, 1, 2, . . . , Ni − 1. Note that N1 N2 N3 = N , the total numberof cells in the crystal. The Ni’s are the number of cells in the

→ai direction. With this

choice*, consider

1

N →k

Σ e−i→k⋅

→Ru →

k ,α =1

N →k

Σ→R′Σ ei

→k⋅(

→R′−

→R)u →

R′,α =→R′Σ u →

R′,α

3

i=1Π

1

Ni

Ni−1

mi=0Σ e2π imi(ni ′−ni)/Ni

.(7.4.12)

The sums in parenthesis are geometric series and give

1

Ni

Ni−1

mi=0Σ e2π imi(ni ′−ni)/Ni =

1

Ni

e2π i(ni ′−ni) − 1

e2π i(ni ′−ni)/Ni − 1

= δ ni ′,ni, (7.4.13)

which when used in Eq. (7.4.12) shows that

*The more common choice for the range of the mi’s is mi = −Ni /2, . . . , 0, . . . , Ni /2, which, cf. Eq.(7.4.11), means that −1/2 ≤ ki ≤ 1/2.

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1

N →k

Σ e−i→k⋅

→Ru →

k ,α = u →R,α . (7.4.14)

The sums over→k are really sums over the mi’s, and thus, ki hardly changes as we go from

mi → mi + 1 for large Ni , cf. Eq. (7.4.11). This allows the sums over mi to be replaced byintegrals; i.e.,

→k

Σ . . . →3

i=1Π ∫

Ni−1

0dmi . . . =

V

(2π )31st Brillouin

zone

∫ ∫ ∫ d→k . . . , (7.4.15)

where V = Nν cell is the volume of the system.

The main goal of this discussion is to obtain the vibrational density of states. By using Eq.(7.4.15), it is easy to show that

g(ω ) =V

(2π )31st Brillouin

zone

∫ ∫ ∫ d→k δ (ω − ω (k)), (7.4.16)

where δ (x) is the Dirac δ -function. Once the frequencies are known, it is relatively easy tonumerically bin them by frequency as a function of k.

7.5. Normal Modes in Crystals: An Example

Consider a crystal with one atom of mass m per unit cell and nearest neighbor interactionsof the type considered in Sec. 7.3. Newton’s equations of motion for this atom becomes

m→∆ ¨0,0,0 = K [ex(∆x

1,0,0 + ∆x−1,0,0 − 2∆x

0,0,0) + ey(∆y0,1,0 + ∆y

0,−1,0 − 2∆y0,0,0)

+ez(∆x0,0,1 + ∆x

0,0,1 − 2∆x0,0,0)], (7.5.1)

where ei is a unit vector in the i direction. Note that this model implies that vibrations in the x, y,z, directions are separable. By using this in Eq. (7.4.5) we see that

Kk = −4ω 20

sin2(k1π )

0

0

0

sin2(k2π )

0

0

0

sin2(k3π )

, (7.5.2)

where ω0 ≡ (K /m)1/2 and where the identity, 1 − cos(x) = 2 sin2(x/2) was used. Since Kk is

already diagonal, we see that ω i(→k) = 2ω0 sin(kiπ ), i = 1, 2, 3, which is also the result for a one

dimensional chain (as was expected, given the separability of the vibrations in the x, y, and z

directions). The normal mode is just ei , which shows that our model potential is too simple.Consider the first eigen-vector. It corresponds to an arbitrary displacement in the x direction,with an eigenvalue that is independent of k y and kz . When k x = 0 we don’t get a single zero fre-quency, as expected, but one for any of the N y Nz values of (k y, kz). Thus, with 3N 2/3 zero

Winter, 2018


frequencies, not 6, the crystal is unstable!

The problem is easily fixed by modifying the interaction potential, i.e., we replace Eq.(7.3.5) by

U ≡K

2(

→∆1 −

→∆2)2, (7.5.3)

which is invariant under rigid translations and rotations, and allows us to rewrite Eq. (7.5.1) as

m→∆0,0,0 = K (

→∆1,0,0 +

→∆−1,0,0 − 2

→∆0,0,0 +

→∆0,1,0 +

→∆0,−1,0 − 2

→∆0,0,0 +

→∆0,0,1 +

→∆0,0,1 − 2

→∆0,0,0). (7.5.4)

With this, it follows that

Kk = −4ω 20[sin2(k1π ) + sin2(k2π ) + sin2(k3π )]

↔1, (7.5.5)

where↔1 is a 3 × 3 identity matrix and where we have expressed

→k in terms of the reciprocal lat-

tice basis vectors, cf. Eq (7.5.11). The triply degenerate vibrational frequencies are simply

ω i(→k) = 2ω0[sin2(k1π ) + sin2(k2π ) + sin2(k3π )]1/2. (7.5.6)

If all the ki → 0, Eq. (7.5.6) becomes ω (→k) ∼ 2ω0kπ , where k ≡ (k2

1 + k22 + k2

3)1/2 which is theexpected linear dispersion law for long wevelength, acoustic vibrations. Some constant fre-quency surfaces in the First Brillouin Zone are shown in Figs. 7.5.1-7.5.3.

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Fig. 7.5.1. A constant normal mode frequency sur-face for ω /ω0 = 1. The reciprocal lattice basis wasused which need not be orthogonal (although it is forthe SCC lattice). Also note that we’ve switched tothe other definition of the first Brillouin zone, with−1/2 ≤ ki ≤ 1/2. The surface is roughly spherical, asexpected for long wav elength acoustic phonons.

Fig. 7.5.2. As in Fig. 7.5.1 but with ω /ω0 = 2.Notice the C4 axises through the centers of any of theunit cell faces.

Fig. 7.5.3. As in Fig. 7.5.1 but withω /ω0 = 3.

Normal mode frequencies for this model were computed numerically for a uniform sample ofki’s in the First Brillouin Zone and binned in order to get the vibrational density of of states asshown in Fig. 7.5.4

Winter, 2018


Fig. 7.5.4. Vibrational density of states for the isotropic model defined byEq. (7.5.3). Any single atom per unit cell lattice will give the same result.The data was obtained by numerically binning the frequencies in the FirstBrillouin Zone. Note that the maximum frequency for this model is 2√3ω0.

By noting that

δ ( f (x)) =iΣ δ (x − xi)

df (xi)

dxi

, (7.5.7)

where f (x) has zeros at x = xi , i = 1, . . ., and that

δ (x) = ∫∞−∞

ds

2πeixs, (7.5.8)

we can rewrite Eq. (7.4.16) as

g(ω )

3N= 2ω ∫

1

0dk1 ∫

1

0dk2 ∫

1

0dk3 δ (ω 2 − ω 2(k))

=ωπ

∞

∫−∞ds ∫

1

0dk1 ∫

1

0dk2 ∫

1

0dk3 eis(ω 2−ω 2(k))

=ωπ

∞

∫−∞ds eisω 2

Φ3(s),

where the extra factor of 3 is due to the triple degeneracy of each mode and where

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Φ(s) ≡ ∫1

0dk e−4iω 2

0 s sin2(π k)

When we use Eq. (7.5.7) for the frequencies, it follows that the wav evector integrations separateand

g(ω ) = 2Vω ∫∞−∞

ds

2πeisω 2

Φ3(s), (7.5.9)

where

Φ(s) ≡ ∫dk1

2πe−is4ω 2

0 sin2(π k1) =e−2ω 2

0is

2π 2 ∫π

0dz e2ω 2

0is cos(z)

=e−2ω 2

0is

2π 2 ∫1

−1dz

e2ω 20isz

(1 − z2)1/2=

e−2ω 20is

2πJ0(2ω 2

0 s), (7.5.10)

where J0(x) is a Bessel function of the first kind.* When this is used in Eq. (7.5.10), we see that

g(ω ) =2Vω(2π )4 ∫

∞−∞

ds eis(ω 2−6ω 20)J3

0 (2ω 20 s) 7.5.11)

*The integral leading to the last equality can be found in I.S. Gradsheyn and I.M. Ryzhik, Table of Inte-

grals, Series, and Products, A. Jeffrey editor, (Academic Press, 1980), Eq. (3.387.2) on p. 321.

Winter, 2018

Chemistry 593 -54- The q → 0 Limit of the Structure Factor

8. The q → 0 Limit of the Structure Factor

For a uniform fluid, we have shown that the structure factor,

NSq = N1 + ρ ∫ dr12 eiq⋅r12[g(r12) − 1]

, (8.1)

where, strictly speaking, g(r) is the pair correlation function for the canonical ensemble. In orderto show how the q → 0 limit comes about, note that the integral in Eq. (8.1) depends mainly ondistances r<∼σ , where σ characterizes the range of the pair correlation function; in most cases,this length is microscopic. Moreover, on this finite length scale, the correlations should be equiv-alent to those in an open system (i.e., in a grand canonical ensemble) since the rest of the liquidcan be viewed as playing the role of a particle reservoir.

In a grand canonical ensemble, the generic pair distribution function, ρ (2)(r1, r2) is definedas the average over the number of particles, N, of the canonical ones; i.e.,

ρ (2)(r1, r2) ≡∞

N=2Σ eβ µN Q(N , V , T )

Ξ(µ, V , T )N (N − 1)

∫ dr3. . . drN e−βU

Zc(N , V , T ). (8.2)

From this definition, it is easy to see that the grand canonical partition function satisfies:

∫ dr1dr2 ρ (2)(r1, r2) = < N (N − 1) > . (8.3)

In order to proceed, we rewrite Eq. (8.1) as:

< N > Sq = < N > + ∫ dr1dr2 eiq⋅r12

ρ (2)(r1, r2) −

< N >

V

2, (8.4)

where it should be remembered that the density of the grand canonical ensemble equals that ofthe canonical one. If we now let q → 0 and use Eq. (8.3), we find that

q→0+lim < N > Sq = < N > + < N (N − 1) > − < N >2

= < (N− < N >)2 >

= < N > kBT ργ P ,

where the isothermal compressibility is defined by

γ P ≡ −1

V

∂V

∂P

N ,T

.

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The q → 0 Limit of the Structure Factor -55- Chemistry 593

What happens to the scattering intensity as the system approaches a critical point? What doesthis imply for g(r)?

Winter, 2018

Chemistry 593 -56- Gaussian Coil Elastic Scattering

9. Gaussian Coil Elastic Scattering

Here is a more exact treatment of elastic scattering from coils with Gaussian segment dis-tributions, that is, where the probability that a pair of monomers, i and j, are separated by a dis-tance R is given by

Pi, j(R) =exp[−R2/2 < R2

i, j >]

[2π < R2i, j >]1/2

, (9.1)

where from the central-limit theorem,

< R2i, j >= |i − j| < l2 > /3 (9.2)

where √ < l2 > is the RMS average bond length (and becomes the bond length in the freely-jointed chain). The structure factor* for the chain is just

S(q) ≡ N −2

i, jΣ ⟨eiq⋅ri, j ⟩, (9.3)

where N is the number of monomers in the polymer. When the distribution given by Eq. (9.1) isused, Eq. (9.3) becomes

S(q) = N −2

i, jΣ e−q2|i− j|<l2>/6 (9.4a)

= N −1 + 2N −2N

i=2Σ

i−1

j=1Σ e−q2(i− j)<l2>/6, (9.4b)

where the sum in Eq. (9.4a) has been split into three parts in order to obtain Eq. (9.4b), i.e., theterm with i = j, i < j, and i > j, where the last two sums are equal. The sum over j in Eq. (9.4b)is simply a geometric series and when summed gives

S(q) = N −1 + 2N −2N

i=2Σ e−q2i<l2>/6 − e−q2<l2>/6

e−q2<l2>/6 − 1(9.5a)

= N −1 + 2N −2

e−q2(N+1)<l2>/6 − e−2q2<l2>/6

(e−q2<l2>/6 − 1)2−

(N − 1)e−q2<l2>/6

e−q2<l2>/6 − 1

(9.5b)

= N −1 1 + e−q2<l2>/6

1 − e−q2<l2>/6+

2N −2e−q2<l2>/6(e−q2 N<l2>/6 − 1)

(e−q2<l2>/6 − 1)2, (9.5c)

*Actually, since inter-chain effects will not be included, this is really the form-factor.

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Gaussian Coil Elastic Scattering -57- Chemistry 593

where (9.5b) is obtained by summing the series in Eq. (9.5a).

In light scattering experiments, q2 < l2 ><< 1. On the other hand, R2 ≡ N < l2 > can belarge (i.e., comparable to the wav elength of light) and hence, qR is not necessarily small. To seehow Eq. (9.5b) simplifies for light scattering, let y ≡ q2 R2/6; this allows Eq. (9.5b) to be rewrit-ten as

S(q) = N −1 1 + e−y/N

1 − e−y/N+

2N −2e−y/N (e−y − 1)

(e−y/N − 1)2. (9.6)

Since N is large, the exponentials can be expanded in Taylor series. When the leading orderterms are kept, we find that

S(q) =2

y2(e−y + y − 1) + O(N −1). (9.7)

Thus, a universal scattering function is obtained for large chains. Note that when y → 0,S(q) → 1, as it must from Eq. (9.3). Finally, S(q) is shown below. The figure also shows theresult for rigid rod-like molecules (which can be treated in a manner analogous to that presentedabove).

Winter, 2018

Chemistry 593 -58- Some Properties of the Master Equation

10. Some Properties of the Master Equation

In the continuum limit the master equation can be written as*

∂Pn(t)

∂t=

n′Σ[Wn′→n Pn′(t) − Wn→n′Pn(t)], (10.1)

where Pn(t) is the probability of finding the system in state n (here assumed discrete) at time t,Wi→ j is the transition probability per unit time for switching from state i to state j, and where thetwo terms in the sum are so-called birth (growth, gain) and death (decay, loss) terms, respec-tively. At present, Wn′→n is non-negative but otherwise arbitrary. In order that Eq. (10.1) makesense, it must conserve probability; i.e.,

nΣ Pn(t) = 1 or

nΣ ∂Pn(t)

∂t= 0. (10.2)

By using Eq. (10.1) in Eq. (10.2) it follows that

n,n′Σ [Wn′→n Pn′(t) − Wn→n′Pn(t)]

must vanish, as is easily shown by exchanging the dummy summation indices in one of theterms.

Next we will show that the master equation has an equilibrium solution, denoted as Peqn .

Clearly, cf. Eq. (10.1), a necessary condition is that

n′Σ

Wn′→n P

eq

n′ − Wn→n′Peqn

= 0, (10.3)

which is satisfied if we impose detailed balance, i.e.,

Wn′→n

Wn→n′=

Peqn

Peq

n′= e−∆En′→n/kBT . (10.4)

The last equality is obtained for a canonical distribution, where ∆En′→n ≡ En − En′ is the energydifference between states n and n′ and is the basis of the Metropolis Monte Carlo method.†

Returning to the general case, we multiply both sides of Eq. (10.3) by arbitrary numbersψ n and sum over n; this gives

0 =n,n′Σ

Wn′→n P

eq

n′ψ n − Wn→n′Peqn ψ n

=n,n′Σ Wn′→n P

eq

n′ (ψ n − ψ n′), (10.5)

*The discussion in this section is based on that of N.G. van Kampen, Stochastic Processes in Physics and

Chemistry, (North-Holland Pub. Co., Amsterdam, 1984).†N. Metropolis, A.W. Metropolis, M.N. Rosenbluth, A.H. Teller, and E. Teller, J. Chem. Phys. 21, 1087(10.1953).

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Some Properties of the Master Equation -59- Chemistry 593

where the last equality is obtained by exchanging the dummy sum variables in the second term.Next, consider the following quantity

H(t) ≡nΣ Peq

n f

Pn(t)

Peqn

, (10.6)

where f (x) is, as yet, an arbitrary function. By using Eq. (10.1) in (10.6) we see that

dH(t)

dt=

n,n′Σ f ′((Pn(t)/Peq

n ))[Wn′→n Pn′(t) − Wn→n′Pn(t)] (10.7a)

=n,n′Σ Wn′→n P

eq

n′ [xn′ f ′(xn) − xn′ f ′(xn′)], (10.7b)

where xn ≡ Pn(t)/Peqn . Finally, let ψ n ≡ f (xn) − xn f ′(xn) and add the result to Eq. (10.7b). This

gives

dH(t)

dt=

n,n′Σ Wn′→n P

eq

n′(xn′ − xn) f ′(xn) + f (xn) − f (xn′)

. (10.8)

Given that Wn′→n and Pn are positive the sign of each term in Eq. (10.8) is determined by thesign of the terms in the ( )’s. These terms are negative, and hence, H(t) is a decreasing function,if we take f (x) to be convex on [0, ∞). For example, let f (x) ≡ x log(x). In this case,

( ) = (xn′ − xn)[log(xn) + 1] + xn log(xn) − xn′ log(xn′) (10.9a)

= xn′log

xn

xn′

+ 1 −xn

xn′

, (10.9b)

which vanishes for xn = xn′ and is otherwise negative.‡ Thus, H(t) will decrease until xn = xn′for all n and n′, i.e., at equilibrium when Pn = Peq

n . Note that we have implicitly assumed thatthere is a unique equilibrium state. Finally, note the similarity between −H(t) and the entropy inequilibrium statistical mechanics; we can make two observations: 1. We’v e shown that there is asecond law; 2. H(t) isn’t unique (what other properties of the entropy would further constrainour choice of f (x)?)

‡This is a consequence of the Klein inequality, namely,

x log x − x log y − x + y ≤ 0,

for x, y∈[0, ∞) and vanishes only for x = y. Prove the Klein inequality.

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Chemistry 593 -60- Critical Phenomena

11. Critical Phenomena

11.1. Introduction

In these lectures some of the general features of the phenomena of phase transitions inmatter will be examined. We will first review some of the experimental phenomena. We thenturn to a discussion of simple thermodynamic and so-called mean-field theoretical approaches tothe problem of phase transitions in general and critical phenomena in particular, showing whatthey get right and what they get wrong. Finally, we will examine modern aspects of the problem,the scaling hypothesis and introduce the ideas behind a renormalization group calculation.

Fig. 11.1. Phase Diagram of Water.1

Fig. 11.2. Liquid-Vapor P-V phase diagramisotherms near the critical point.2

Consider the two well known phase diagrams shown in Figs. 11.1 and 11.2. Along any ofthe coexistence lines, thermodynamics requires that the chemical potentials in the coexistingphases be equal, and this in turn gives the well known Clapeyron equation:

dP

dT

coexistence

=∆H

T ∆V, (11.1.1)

where ∆H and ∆V are molar enthalpy and volume changes, respectively, and T is the tempera-ture. Many of the qualitative features of a phase diagram can be understood simply by using theClapeyron equation, and knowing the relative magnitudes and signs of the enthalpy and volumechanges. Nonetheless, there are points on the phase diagram where the Clapeyron equation can-not be applied naively, namely at the critical point where ∆V vanishes.

1G. W. Castellan, Physical Chemistry, 3rd ed., (Benjamin Pub. Co., 1983), p. 266.2R.J. Silbey and R.A. Alberty, Physical Chemistry, 3rd ed., (John Wiley & Sons, Inc. 2001) p. 16.

Winter, 2018

Critical Phenomena -61- Chemistry 593

The existence of critical points was controversial when it was first considered in the 19thcentury because it means that you can continuously transform a material from one phase (e.g., aliquid) into another (e.g., a gas). We now hav e many experimental examples of systems thathave critical points in their phase diagrams; some of these are shown in Table 11.1. In each case,the nature of the transition is clearly quite different (from the point of view of the qualitativesymmetries of the phases involved).

TABLE 11.1. Examples of critical points and their order parameters3

Critical OrderPoint Parameter

Example Tc(oK )

Liquid-gas Density H2O 647.05

Ferromagnetic Magnetization Fe 1044.0

Anti-ferromagnetic Sub-lattice FeF2 78.26magnetization

Super-fluid 4 He-amplitude 4 He 1.8-2.1

Super- Electron pair Pb 7.19conductivity amplitude

Binary fluid Concentration CCl4-C7F14 301.78mixture of one fluid

Binary alloy Density of one Cu − Zn 739kind on a sub-lattice

Ferroelectric Polarization Triglycine 322.5sulfate

The cases in Table 11.I are examples of so-called 2nd order phase transitions, according tothe naming scheme introduced by P. Eherenfest. More generally, an nth order phase transition isone where, in addition to the free energies, (n − 1) derivatives of the free energies are continuousat the transition. Since the first derivatives of the free energy give entropy and volume, all of thefreezing and sublimation, and most of the liquid-vapor line would be classified as first-order tran-sition lines; only at the critical point does it become second order. Also note that not all phasetransitions can be second order; in some cases, symmetry demands that the transition be firstorder.

At a second order phase transition, we continuously go from one phase to another. Whatdifferentiates being in a liquid or gas phase? Clearly, both have the same symmetries, so whatquantitative measurement would tell us which phase we are in? We will call this quantity (or

3S.K. Ma, Modern Theory of Critical Phenomena, (W.A. Benjamin, Inc., 1976), p. 6.

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quantities) an order parameter, and adopt the convention that it is zero in the one phase region ofthe phase diagram. In some cases there is a symmetry difference between the phases and thismakes the identification of the order parameters simpler, in others, there is no obvious uniquechoice, although we will show later that for many questions, the choice doesn’t matter.

For example, at the liquid-gas critical point the density (molar volume) difference betweenthe liquid and vapor phases vanishes, cf. Fig. 11.2, and the density difference between the twophases is often used as the order parameter. Second order transitions are also observed in ferro-magnetic or ferroelectric materials, where the magnetization (degree or spin alignment) or polar-ization (degree of dipole moment alignment) continuously vanishes as the critical point isapproached, and we will use these, respectively, as the order parameters. Other examples aregiven in Table 11.1.

At a second order critical point, many quantities vanish (e.g., the order parameter) whileothers can diverge (e.g., the isothermal compressibility, −V −1(∂V /∂P)T ,N cf. Fig. 11.2). In orderto quantify this behavior, we introduce the idea of a critical exponent. For example, consider aferromagnetic system. As we just mentioned, the magnetization vanishes at the critical point(here, this means at the critical temperature and in the absence of any externally applied mag-netic field, H), thus near the critical point we might expect that the magnetization, m might van-ish like

m∝|Tc − T |β , when H = 0, (11.1.2)

or at the critical temperature, in the presence of a magnetic field,

m∝H1/δ . (11.1.3)

The exponents β and δ are examples of critical exponents and are sometimes referred to as theorder parameter and equation of state exponents, respectively; we expect both of these to be posi-tive. Other thermodynamic quantities have their own exponents; for example, the constant mag-netic field heat capacity (or CP in the liquid-gas system) can be written as

CH ∝|Tc − T |−α , (11.1.4)

while the magnetic susceptibility, χ , (analogous to the compressibility) becomes

χ ∝|Tc − T |−γ . (11.1.5)

Non-thermodynamic quantities can also exhibit critical behavior similar to Eqs.(11.1.2)−(11.1.5). Perhaps the most important of these is the scattering intensity measured inlight or neutron scattering experiments. As you learned in statistical mechanics (or will see againlater in this course), the elastic scattering intensity at scattering wav e-vector q is proportional tothe static structure factor

NS(q) ≡< |N (q)|2 > , (11.1.6)

where N (q) is the spatial Fourier transform of the density (or magnetization density), and < . . . >denotes an average in the grand canonical ensemble. In general, the susceptibility or compress-ibility and the q → 0 limit of the structure factor4 are proportional, and thus, we expect the

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scattered intensity to diverge with exponent γ as the critical point, cf. Eq. (11.1.5). This isindeed observed in the phenomena called critical opalescence. At T = Tc and non-zero wav e-vectors, we write,

S(q)∝1

q2−η . (11.1.7)

Finally, we introduce one last exponent, one that characterizes the range of molecular cor-relations in our systems. The correlation-length is called ξ , and we expect that

ξ ∝|Tc − T |−ν , (11.1.8)

where we shall see later that the correlation-length exponent, ν > 0.

Some experimental values for these exponents for ferromagnets are given in Table 11.2.The primes on the exponents denote measurements approaching the critical point from the two-phase region (in principle, different values could be observed). What is interesting, is that eventhough the materials are comprised of different atoms, have different symmetries and transitiontemperatures, the same critical exponents are observed, to within the experimental uncertainty.

TABLE 11.2. Exponents at ferromagnetic critical points5

Material Symmetry T (oK ) α , α ′ β γ , γ ′ δ ηFe Isotropic 1044.0 α = α ′ = 0. 120 0.34 1.333 0.07

±0.01 ±0.02 ±0.015 ±0.07

Ni Isotropic 631.58 α = α ′ = 0. 10 0.33 1.32 4.2±0.03 ±0.03 ±0.02 ±0.1

EuO Isotropic 69.33 α = α ′ = 0. 09±0.01

YFeO3 Uniaxial 643 0.354 γ = 1. 33±0.005 ±0.04

γ ′ = 0. 7±0.1

Gd Anisotropic 292.5 γ = 1. 33 4.0±0.1

Of course, this behavior might not be unexpected. After all, these are all ferromagnetictransitions; a phase transition where "all" that happens is that the spins align. What is more inter-esting are the examples shown in Table 11.3. Clearly, the phase transitions are very differentphysically; nonetheless, universal values for the critical exponents seem to emerge.

4See, e.g., http://ronispc.chem.mcgill.ca/ronis/chem593/structure_factor.1.html.5S.K. Ma, op. cit., p. 12.

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TABLE 11.3. Exponents for various critical points6

Critical

PointsMaterial Symmetry Tc(oK ) α , α ′ β γ , γ ′ δ η

Antiferro- CoCl2 ⋅ 6H2O Uniaxial 2.29 α ≤ 0. 11 0.23

magnetic α ′ ≤ 0. 19 ± 0. 02

FeF2 Uniaxial 78.26 α = α ′ = 0. 112

± 0. 044

RbMnF3 Isotropic 83.05 α = α ′ = −0. 139 0.316 γ = 1. 397 0.067

± 0. 007 ± 0. 008 ± 0. 034 ± 0. 01

Liquid-gas CO2 n = 1 304.16 α ∼1/8 0.3447 γ = γ ′ = 1. 20 4.2

± 0. 0007 ± 0. 02

Xe 289.74 α = α ′ = 0. 08 0.344 γ = γ ′ = 1. 203 4.4

± 0. 02 ± 0. 003 ± 0. 002 ±0. 4

3 He 3.3105 α ≤ 0. 3 0.361 γ = γ ′ = 1. 15

α ′ ≤ 0. 2 ±0. 001 ±0. 03

4 He 5.1885 α = 0. 127 0.3554 γ = γ ′ = 1. 17

α ′ = 0. 159 ±0. 0028 ±0. 0005

Super-fluid 4 He 1.8-2.1 0. 04 ≤ α = α ′ < 0

Binary CCl4 − C7F14 n = 1 301.78 0.335 γ = 1. 2 ∼ 4

Mixture ±0. 02

Binary Co − Zn n = 1 739 0.305 γ = 1. 25

alloy ±0. 005 ±0. 02

Ferro- Triglycine n = 1 322.6 γ = γ ′ = 1. 00

electric sulfate ±0. 05

Our goals in these lectures are as follows:

1. To come up with some simple theory that results in phase transitions in general, and sec-ond order phase transitions in particular.

2. To show how universal critical exponents result.

3. To be able to predict the correct values for the critical exponents.

It turns out the 1. and 2. are relatively easily accomplished; 3. is not and Kenneth G. Wilson, wonthe 1982 physics Nobel Prize for showing how to calculate the critical exponents.

11.2. Thermodynamic Approach

11.2.1. General Considerations

Other than the already mentioned Clapeyron equation, cf. Eq. (11.1.1), and its generaliza-tions to higher order phase transitions (not discussed), thermodynamics has relatively little to sayabout the critical exponents. One class of inequalities can be obtained by using thermodynamicstability requirements (e.g., that arise by requirements that the free energy be a minimum at

6S.K. Ma, op. cit., pp. 24-25.

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equilibrium). As an example of how this works, recall the well known relationship between theheat capacities CP and CV , namely,

CP = CV +TVγ 2

T

γ P

, (11.2.1)

where γT ≡ V −1(∂V /∂T )P,N is the thermal expansion coefficient and γ P ≡ −V −1(∂V /∂P)T ,N is theisothermal compressibility. Since thermodynamic stability requires that CV and γ P be positive, itfollows that

CP ≥TVγ 2

T

γ P

. (11.2.2)

By using the different exponent expressions, Eqs. (11.1.2) (11.1.4) and (11.1.5), this last inequal-ity implies that, as T → Tc,

τ −α ≥ positive constant × τ 2(β −1)+γ , (11.2.3)

where τ ≡ |T − Tc |/Tc. The inequality will hold at Tc only if

α + 2β + γ ≥ 2. (11.2.4)

This is known as the Rushbrook inequality. If you check some of the experimental data given inTables S2 and 11.3, you will see that in most of the cases, α + 2β + γ ≈ 2, and to within theexperimental error, the inequality becomes an equality. This is no accident!

11.2.2. Landau-Ginzburg Free Energy

We now try to come up with the simplest model for a free energy or equation of state thatcaptures some of the physical phenomena introduced above. For example, we could analyze thewell known van der Waals equation near the critical point. It turns out however, that a modelproposed by Landau and Ginzburg is even simpler and in a very general manner shows many ofthe features of systems near their critical points. Specifically, they modeled free energy differ-ence between the ordered and disordered phases as

∆G ≡ −HΨ +A

2Ψ2 +

B

3Ψ3 +

C

4Ψ4+. . . , (11.2.5)

where A, B, C, etc., depend on the material and on temperature, and where H plays the role ofan external field (e.g., magnetic or electric or pressure).

In some cases, symmetry can be used to eliminate some of the terms in ∆G; for example,in systems with inversion or reflection symmetry (magnets), in the absence of an external fieldeither Ψ or −Ψ must give the same free energy. This means that the free energy must be an evenfunction of Ψ in the absence of an external field, and from Eq. (11.2.5) we see that this impliesthat B = 0. Examples of the Landau free energy for ferromagnets are shown in Figs. 11.3 and11.4.

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Fig. 11.3. The Landau-Ginzberg free energy (cf. Eq.(11.2.5)) for ferromagnets (B = 0. 0) at zero externalmagnetic field, and C = 1. 0.

Fig. 11.4. The Landau-Ginzberg free energy, cf. Eq.(11.2.5), for ferromagnets (B = 0. 0) at non-zeroexternal magnetic field (H = 1. 0), and C = 1. 0.

The minima of the free energy correspond to the stable and metastable thermodynamicequilibrium states. In general, we see that an external field induces order (i.e., the free energyhas a minimum with Ψ ≠ 0 when H ≠ 0) and that multiple minima occur for A < 0. When theexternal field is zero, there are a pair of degenerate minima when A < 0. This is like the behaviorseen at the critical point, where we go from a one- to two-phase region of the phase diagram, cf.Fig. 11.2. To make this more quantitative, we assume that

A∝T − Tc, as T → Tc, (11.2.7)

with a positive proportionality constant, while the other parameters are assumed to be roughlyconstant in temperature near Tc.

In order to extract the critical exponents, the equilibrium must be analyzed more carefully.The equilibrium state minimizes the free energy, and hence, Eq. (11.2.5) gives:

H = AΨ + BΨ2 + CΨ3. (11.2.8)

For ferromagnets with no external field, B = 0, and Eq. (11.2.8) is easily solved, giving

Ψ = 0 (11.2.9a)

and

Ψ = ±√ −A

C. (11.2.9b)

Clearly, the latter makes physical sense only if A < 0, i.e., according to the preceding discussion,when T < Tc. Indeed, for A < 0 the it is easy to see that the nonzero roots correspond to the min-ima shown in Fig. 11.3, while Ψ = 0 is just the maximum separating them, and is thus not the

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equilibrium state.

With the assumed temperature dependence of A, cf. Eq. (11.2.7), we can easily obtain thethe critical exponents. For example, from Eqs. (11.2.7) and (11.2.9b), it follows thatΨ∝(Tc − T )1/2, and thus, β = 1/2. In the absence of a magnetic field, the free energy differencein the equilibrium state is easily shown to be

∆G =

0, when A > 0 (T > Tc)

−A2

4C, otherwise.

(11.2.10)

Since

CH = T

∂S

∂T

H ,N

= −T

∂2G

∂T 2

H ,N

, (11.2.11)

it follows that the critical contribution to the heat capacity is independent of temperature, andhence, α = α ′ = 0.

An equation for the susceptibility can be obtained by differentiating both sides of Eq.(11.2.8) with respect to magnetic field and solving for χ ≡ (∂Ψ/∂H)T ,H=0. This gives:

χ =1

A + 2BΨ + 3CΨ2=

1

A, for T > Tc

1

2|A|, for T < Tc,

(11.2.12)

which shows, cf. Eq. (11.1.5), that γ = γ ′ = 1, and also shows that the amplitude of the diver-gence of the susceptibility is different above and below Tc.

Finally, by comparing Eq. (11.2.8) at T = Tc (A = 0) with Eq. (11.1.3) we see that δ = 3.These results are summarized in Table 11.4. Note that the Rushbrook inequality is satisfied as anequality, cf. Eq. (11.2.4).

Table 11.4. Mean-Field Critical Exponents

Quantity Exponent Value

Heat Capacity α 0Order Parameter β 1/2Susceptibility γ 1Eq. of State at Tc δ 3Correlation length ν 1/2Correlation function η 0

The table also shows the results for the exponents η and ν , which strictly speaking, don’t arisefrom our simple analysis. They can be obtained from a slightly more complicated version of thefree energy we’ve just discussed, one that allows for thermal fluctuations and spatially

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nonuniform states. This is beyond the scope of present discussion and will not be pursued fur-ther here.

Where do we stand? The good news is that this simple analysis predicts universal valuesfor the critical exponents. We’ve found values for them independent of the material parameters.Unfortunately, while they are in the right ball-park compared to what is seen experimentally, theyare all quantitatively incorrect. In addition, the Landau-Ginzburg model is completely phe-nomenological and sheds no light on the physical or microscopic origin of the phase transition.

11.3. Weiss Mean-Field Theory

The first microscopic approach to phase transitions was given by Weiss for ferromagnets.As is well known, a spin in an external magnetic field has a Zeeman energy given by

E = −γ h− H ⋅ S, (11.3.1)

where S is the spin operator, γ is called the gyromagnetic ratio, and H is the magnetic field at thespin (which we use to define the z axis of our system).

First consider a system of non-interacting spins in an external field. This is a simple prob-lem in statistical thermodynamics. If the total spin is S, the molecular partition function, q, isgiven by

q =S

Sz=−SΣ eα Sz =

sinh[α (S + 1/2)]

sinh(α /2), (11.3.2)

where α ≡ γ h− H /(kBT ), kB is Boltzmann’s constant, and where the second equality is obtainedby realizing that the sum is just a geometric series.

With the partition function in hand it is straightforward, albeit messy, to work out variousthermodynamic quantities. For example, the average spin per atom, < s >, is easily shown to begiven by

< s >=∂ ln q

∂α= BS(α ), (11.3.3)

where

BS(α ) ≡ (S + 1/2)coth[α (S + 1/2)] − coth(α /2)/2 (11.3.4)

is known as the Brillouin function. The average energy per spin is just −γ h− H < s >, while theHelmholtz free energy per spin is −kBT ln q, as usual. The spin contribution to the heat capacityis obtained by taking the temperature derivative of the energy and becomes:

CH

NkB

= α 2

1

4 sinh2(α /2)−

(S + 1/2)2

sinh2[(S + 1/2)α ]

. (11.3.5)

Other thermodynamic quantities are obtained in a similar manner. The magnetization and spincontributions to the heat capacity are shown in Figs. 11.5 and 11.6.

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Fig. 11.5. Spin polarization of an ideal spin in anexternal magnetic field.

Fig. 11.6. Spin contribution to the constant magneticfield heat capacity, CH .

While this simple model gets many aspects of a spin system correct (e.g., the saturationvalues of the magnetization and the high temperature behavior of the magnetic susceptibilities),it clearly doesn’t describe any phase transition. The magnetization vanishes when the field isturned off and the susceptibility is finite at any finite temperature. Of course, the model didn’tinclude interactions between the spins, so no ordered phase should arise.

Weiss included magnetic interactions between the spins by realizing that the magnetic fieldwas made up of two parts: the external magnetic field and a local field that is the net magneticfield associated with the spins on the atoms surrounding the spin in question. In a disorderedsystem (i.e., one with T > Tc and no applied field) the neighboring spins are more or less ran-domly oriented and the resulting net field vanishes, on the other hand, in a spin aligned systemthe neighboring spins are ordered and the net field won’t cancel out. To be more specific, Weissassumed that

H = Hext + λ < s > , (11.3.6)

where Hext is the externally applied field and λ is a parameter that mainly depends on the crystallattice. In ferromagnets the field of the neighboring atoms tends to further polarize the spin, andthus, λ > 0 (it is negative in anti-ferromagnetic materials). Note that the mean field that goesinto the partition function depends on the average order parameter, which must be determinedself-consistently.

When Weiss’s expression for the magnetic field is used in Eqs. (11.3.3) and (11.3.4) a tran-scendental equation is obtained, i.e.,

< s >= BS((γ h− (Hext + λ < s >) / (kBT ))). (11.3.7)

In general, while it is easy to show that there are at most three real solutions and a critical point,Eq. (11.3.7) must be solved graphically or numerically. Nonetheless, it can be analyzed analyti-cally close to the critical point since there < s > and Hext are small as is α . We can use this bynoting the Taylor series expansion,

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coth(x) =1

x+

x

3−

x3

45+. . . , (11.3.8)

which when used in Eq. (11.3.7) gives

< s >=α3

[(S + 1/2)2 − (1 / 2)2] −α 3

45[(S + 1/2)4 − (1 / 2)4]+. . . . (11.3.9)

If the higher order terms are omitted, Eq. (11.3.9) is easily solved. For example, when Hext = 0,we see that in addition to the root < s >= 0, we have:

< s >= ±√ 45T 2(Tc − T )

[(S + 1/2)4 − (1 / 2)4](γ h− λ /kB)3, (11.3.10)

where the critical temperature (known as the Curie temperature in ferromagnets) is

Tc ≡γ h− λ S(S + 1)

3kB

. (11.3.11)

When T < Tc the state with the nonzero value of < s > has the lower free energy. Thus we’vebeen able to show that the Weiss theory has a critical point and have come up with a microscopicexpression for the critical temperature. By repeating the analysis of the preceding section, onecan easily obtain expressions for the other common thermodynamic functions.

The Weiss mean field theory is the simplest theory of ferro-magnetism, and over the yearsmany refinements to the approach have been proposed that better estimate the critical tempera-ture. Unfortunately, they all fail in one key prediction, namely, the critical exponents are exactlythe same as those obtained in preceding section, e.g., compare Eqs. (11.2.9b) and (11.3.10). Thisshouldn’t be too surprising, given the similarity between Eqs. (11.2.8) and (11.3.9), and thus,while we’ve been able to answer some of our questions, the matter of the critical exponents stillremains.

11.4. The Scaling Hypothesis

When introducing the critical exponents, cf. Table 11.4, we mentioned the exponent νassociated with the correlation length, that is ξ ∼|T − Tc |−ν . What exactly does a diverging corre-lation length mean? Basically, it is the length over which the order parameter is strongly corre-lated; for example, in a ferromagnet above its Curie temperature, if we find a part of the samplewhere the spins are aligned and pointing up, then it is very likely that all the neighboring spinsout to a distance ξ will have the same alignment.

In the disordered phase, far from the critical point the correlation-length is microscopic,typically a few molecular diameters in size. At these scales, all of the molecular details areimportant. What happens as we approach the critical point and the correlation length grows?

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Fig. 11.7. A snapshot of the spin configuration in a computer simulation of the 2D Isingmodel of a ferromagnet slightly above its critical temperature. Dark and light regionscorrespond to spin down and spin up, respectively.

Figure 11.7 shows the spin configuration obtained from a Monte Carlo simulation of theIsing spin system (S = 1/2 with nearest-neighbor interactions) close to its critical point. We seelarge interconnected domains of spin up and spin down, each containing roughly 103 − 104 spins.If this is the case more generally, what determines the free energy and other thermodynamicquantities? Clearly, two very different contributions will arise. One is associated with the short-range interactions between the aligned spins within any giv en domain, while the other involvesthe interactions between the ever larger (as T → Tc) aligned domains. The former shouldbecome roughly independent of temperature once the correlation length is much larger than themolecular lengths and should not contain any of the singularities characteristic of the criticalpoint. The latter, then, is responsible for the critical phenomena and describes the interactionsbetween large aligned domains. As such, it shouldn’t depend strongly on the microscopic detailsof the interactions, and universal behavior should be observed.

The next question is how do these observations help us determine the structure of thequantities measured in thermodynamic or scattering experiments? First consider the scatteringintensity or structure factor, S(q), introduced in Eq. (11.1.6). The scattering wav e-vector, q,

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probes the length-scales present in the density or magnetization fluctuations. From the discus-sion of the preceding paragraph, the only length scale that is relevant near the critical point (atleast for quantities that exhibit critical behavior) is the correlation length ξ ; hence, we should beable to write

S(q, T )∝ξ γ /ν F(qξ ), (11.4.1)

where the factor of ξ γ /ν was introduced in order to capture the divergence in the scattering inten-sity at q = 0 associated with the susceptibility, cf. Eqs. (11.1.5) and (11.1.8). The function F(x)is arbitrary, except for two properties: 1) F(0) is nonzero; and 2) F(x)∼1/x2−η as x → ∞. Theformer implies that there is a nonzero susceptibility, while the latter is necessary if the behaviorgiven in Eq. (11.1.7) is to be recovered. Strictly speaking, Eq. (11.1.7) holds only at the criticalpoint where ξ is infinite, and hence, the factors of ξ must cancel in Eq. (11.4.1); this only hap-pens if

γ = ν (2 − η). (11.4.2)

This sort of relationship between the exponents is known as a scaling law, and seems to hold towithin the experimental accuracy of the measurements.

Widom7 formalized these ideas by assuming that the critical parts of the thermodynamicfunctions were generalized homogeneous functions. For example, for the critical part of themolar free energy, a function of temperature and external field, this means that

G(λ pτ , λ q H) = λG(τ , H) (11.4.3)

for any λ , and where recall that τ ≡ (T − Tc)/Tc. All the remaining critical exponents can begiven in terms of p and q.

We showed in Eq. (11.2.11) that the heat capacity is obtained from two temperature deriva-tives of the free energy. From Eq. (11.4.3) this implies that at H = 0,

λ2pCH (λ pτ ) = λCH (τ ). (11.4.4a)

Since λ is arbitrary, we set it to τ −1/ p and rewrite Eq. (11.4.4a) as

CH (τ ) = τ −(2 p−1) / pCH (1), (11.4.4b)

which gives

α = 2 −1

p. (11.4.5)

Similarly, the magnetization is obtained by taking the derivative of the free energy with respect toH . Thus, Eq. (11.4.3) gives

M(τ , H) = λ q−1 M(λ pτ , λ q H). (11.4.6a)

7B. Widom, J. Chem. Phys., 43, 3898 (1965).

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When H = 0 we set λ = τ −1/ p, as before, and find that

M = τ (1−q)/ p M(1, 0), (11.4.6b)

or

β =1 − q

p. (11.4.7)

At the critical temperature (τ = 0) we let λ = H−1/q, and rewrite Eq.(11.4.6a) as

M = H (1−q)/q M(0, 1), (11.4.8)

giving

δ =q

1 − q. (11.4.9)

Finally, the susceptibility is obtained from the derivative of the magnetization with respect tofield. By repeating the steps leading to the exponent β , we can easily show that

γ =2q − 1

p. (11.4.10)

All four exponents, α , β , δ and γ , hav e been expressed in terms of p and q, and thus twoscaling laws can be obtained. For example, by using Eqs. (11.4.7), (11.4.9) and (11.4.10) it fol-lows that

γ = β (δ − 1), (11.4.11)

while by using Eqs. (11.4.5), (11.4.7) and (11.4.10) we recover the Rushbrook inequality (as anequality), cf. Eq. (11.2.4).

11.5. Kadanoff Transformation and The Renormalization Group

The discussion of the scaling hypothesis given in the preceding section is ad hoc to say theleast. Moreover, even if it is correct, it still doesn’t tell us how to calculate the independent expo-nents, p and q. Kadanoff8 has given a very physical interpretation of what scaling really means,and has shown how to apply it to the remaining problem.

In order to introduce the ideas, consider the Hamiltonian for a ferromagnet:

H = −J<n.n.>Σ si s j − H

iΣ si , (11.5.1)

where < n. n. > denotes a sum over nearest neighbor pairs on the lattice and si ≡ ±1 is a spin vari-able (scaled perhaps by 2) for the atom on the i’th lattice site. This is known as the Ising modeland, with appropriate reinterpretations of the spin variables, can be used to model liquids (e.g.,si = ±1 for empty or filled sites, respectively) solutions, surface adsorption, polymers etc. It can

8L. Kadanoff, Physics 2, 263 (1966).

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also be generalized to allow for more complicated interactions (e.g., between triplets of spins ornon-nearest-neighbors) or to allow for more states per site. Note that J > 0 favors alignment(ferromagnetic order).

For a system of N spins, the exact canonical partition function, Q, is

Q =1

s1=−1Σ

1

s2=−1Σ . . .

1

sN =−1Σ e−H /kBT (11.5.2)

In general, the sums cannot be performed exactly; nonetheless, consider what happens if we wereto split up them up in the following way:

1. Divide up the crystal into blocks, each containing Ld spins (d is the dimension of space),cf Fig. 11.8

2. Fix each block’s spin. There is no unique way to do this. Usually we fix the total spin ofthe block, i.e.,

SL ≡ Zi∈block

Σ Si (11.5.3)

where Z ≈ 1/Ld is introduced to make SL ≈ ±1. Alternately, for Ld odd, we could assignSL ≡ ±1 depending on whether the majority of the spins in the block had spin ±1. As longas the block size is comparable to or smaller than the correlation length, these two choicesshould give the same answer (why?).

3. Average over the internal configurations of each block and calculate the mean interactionpotential between different blocks.

Fig. 11.8. An example of the block transformation on a square lattice. Here L = 2.

With these, the partition function can be rewritten as

Q =SL

Σ e−W /kBT , (11.5.4)

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where

e−W /kBT ≡1

si=−1Σ ′e−H /kBT , (11.5.5)

and where the prime on the spin sums means to only include those configurations that are consis-tent with the block-spin configuration being summed. The effective potential, W , is analogous tothe potential of mean force encountered in statistical mechanics and is just the reversible workneeded to bring the system into a configuration given by the SL’s.

What will W look like as a function of the block spin configuration? Clearly it toodescribes the interactions between spins (now blocks of spins) and should look something likethe original spin Hamiltonian introduced in Eq. (11.5.1), perhaps with some of the additionalterms discussed above. Thus, we expect that

W = −JL<n.n. blocks>

Σ Si S j − HLiΣ Si+. . . , (11.5.6)

where . . . represents the extra terms, and where note that the coupling constant and magneticfield have changed.

Equation (11.5.6) is just the Hamiltonian for a spin system with N /Ld spins. Hence, if weuse it to carry out the remaining sums in Eq. (11.5.4) we see that the original Helmholtz freeenergy, −kBT ln Q is equal to the Helmholtz free energy of a system with fewer spins and differ-ent values for the parameters appearing in the Hamiltonian; nonetheless, it is still the free energyof a spin system and we conclude that the free energy per spin,

A(J , H , . . . ) = L−d A(JL , HL , . . . ), (11.5.7)

where . . . denotes the parameters that appear in the extra terms.

A key issue is to understand what the block transformation does to the parameters in theHamiltonian. If we denote the latter by a column vector µ then our procedure allows us to write

µL = RL(µ) (11.5.8)

and Eq. (11.5.7) becomes

A(µ) = L−d A((RL(µ))). (11.5.9)

Obviously we could have done the block transformation in more than one step, and hence,

RL RL′ = RLL′, (11.5.10)

which some of you may realize is an operator multiplication rule, and has led to the characteriza-tion of the entire procedure as a group called the renormalization-group (RG). (Actually it is onlya semi-group since the inverse operations don’t exist).

In general, the renormalized problem will appear less critical, since the correlation lengthwill be smaller on the re-blocked lattice (remember, we’re simply playing games with how wecarry out the sums, the real system is the same). One exception to this observation is at the

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critical point, where the correlation length is infinite to begin with. In order that the renormal-ized appear as critical as the original one, the Hamiltonians before and after the block transfor-mation must describe critical systems, or equivalently, the renormalized parameters will turn outto be the same as the original critical ones; i.e.,

µ = RL(µ). (11.5.11)

This is known as the fixed point of the RG transformation, and we will denote the special valuesof the parameters at the fixed point as µ*.

Suppose we’re near the critical point and we write µ = µ * +δ µ, where δ µ is not too large.By using Eqs. (11.5.8) and (11.5.11) we can write

δ µL = RL(µ* + δ µ) − RL(µ*) ≈ KLδ µ, (11.5.12)

where

KL ≡

∂RL(µ)

∂µ

µ=µ*

(11.5.13)

is a matrix that characterizes the linearized RG transformation at the fixed point.

Rather than use the parameters directly, it is useful to rewrite δ µ in terms of the normalizedeigenvectors of the matrix KL . These are defined by

KLui = λ i(L)ui , (11.5.14)

with ui ⋅ ui = 1, as usual. It turns out that the eigenvalues must have a very simple dependenceon L. From the multiplication property of the RG transformation, cf. Eq. (11.5.10), we see that

λ i(LL′) = λ i(L)λ i(L′), (11.5.15)

and in turn this implies that

λ i(L) = L yi , (11.5.16)

where the exponent is obtained from the eigenvalue as

yi =ln((λ i(L)))

ln(L). (11.5.17)

We use this eigenanalysis by writing

δ µ =iΣ ciui , (11.5.18)

where we’ve assumed that the ui’s form a basis, and thus the ci’s linear combinations of the δ µ’s.Clearly, the ci’s are just as good at describing the Hamiltonian as the original µ’s. If the RGtransformation is applied to δ µ, and the result expressed in terms of the ci’s, cf. Eqs. (11.5.12),(11.5.14), and (11.5.16), it follows that

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ci,L = L yi ci , (11.5.19)

which no longer involves matrices and has a very simple dependence on L. Indeed, if we goback to our discussion of the free energy associated with the RG transformation, cf. Eq. (11.5.7),and express the parameters in terms of the ci’s, we see that

A(c1, c2, . . . ) = L−d A(L y1 c1, L y2 c2, . . . ), (11.5.20)

which is a generalization of the scaling form assumed by Widom, cf. Eq. (11.4.3); moreover, wecan repeat the analysis of the preceding section to express the experimental exponents in terms ofthe yi’s. Before doing so, however, it is useful to look at some of the qualitative properties of Eq.(11.5.20). The notation is slightly different, but nonetheless, the scaling analysis given above canbe repeated, and the results are summarized in Table 11.5.

Table 11.5: Critical exponents in terms of the yi’s†

Exponent In terms of the yi’s Numerical Value*

α (2y1 − d)/y1 - 0.2671β (d − y2)/y1 0.7152γ (2y2 − d)/y1 0.8367δ y2/(d − y2) 2.1699ν 1/y1 1.1335

†We’v e assumed that c1∝τ and that c2∝H .

*These are for the analysis of the 2d triangular lattice ferromagnet presented in the next

section.

First, as was noted above, the RG transformation may not exactly preserve the form of themicroscopic Hamiltonian, e.g., Eq. (11.5.1); as such, there will be more than two parameters gen-erated by the RG iterations. How does this agree with the thermodynamics, which says that thecritical point in a ferromagnetic or liquid-gas system is determined solely by temperature (cou-pling constant) and magnetic field? What about the other parameters? Whatever they are, it isinconceivable that all phase transitions of the same universality class and materials have the samevalues for these, and thus, how can universal exponents arise? The way out of this problem is forthe other parameters to have neg ative yi exponents. If this is the case, then the scaling analysis(where we write L in terms of the reduced temperature or magnetic field) gives L → ∞ as thecritical point is approached, and these variables naturally assume their fixed point values, i.e.,δ ci → 0. These kind of quantities are called irrelevant variables, since they will adopt theirfixed-point values irrespective of their initial ones. In other words, irrespective of the actualparameter values, as determined by the microscopic nature of the material and phases under con-sideration, close enough to the critical point the systems will all behave like the one with theparameters set to those that characterize the fixed point.

Second, quantities that have positive exponents are called relevant variables, and describethings like temperature and magnetic field. We know that different materials have different criti-cal temperatures, pressures etc., and thus we expect that their values are important. Explicit cal-culations show that only two relevant variables arise for the class of problems under discussion,and so the thermodynamics of our model is consistent with experiment.

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Taken together, these two observations explain why universal behavior is observed at thecritical point and how scaling laws arise. Moreover, we hav e the blueprint for the calculation ofthe critical exponents. All we have to do is to compute the eigenvalues of the linearized RG

transformation and carry out some simple algebra. As we will now see, this isn’t as easy as itsounds.

11.6. An Example

As the simplest (although not very accurate) example9 of the RG approach consider thetwo dimensional triangular lattice depicted in Fig. 11.9.

Fig. 11.9. A portion of a triangular lattice with blocking scheme withL = √3 as indicated. The arrows show the spins that interact on anadjacent pair of blocks.

We will perform the block transformation in blocks of three as indicated (L = √3) using a major-ity rule to assign the block spin. Each block of three can assume 8 spin configurations, 4 willhave the majority spin up (↑↑↑, ↑↑↓, ↑↓↑, ↓↑↑) and 4 spin down (↓↓↓, ↓↓↑, ↓↑↓, ↑↓↓).

It turns out to be difficult to evaluate the restricted sum in Eq. (11.5.5), and we will useperturbation theory to get an approximate expression; specifically, we will treat the interactionsbetween the blocks and the external field as a perturbation. Hence, to leading order the blocksare uncoupled and the partition function can be easily evaluated by explicitly summing the con-figurations. This gives:

e−W (0)

= (e3J + 3e−J )N /3, (11.6.1)

where we absorb the factors of kBT into J and H . Since our answer doesn’t depend on the val-ues of the block spins or the magnetic field, it’s not particularly interesting.

9Th. Neimeijer and J.M.J. van Leeuwen, Phys. Rev. Lett. 31, 1411 (1973); Physica 71, 17 (1974).

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By expanding W and the Hamiltonian in the perturbing terms and comparing the results,it’s easy to show that to first order,10

W (1) =< H (1) >0 . (11.6.2)

When the average is performed we find terms that are linear in the block spin variables, the coef-ficient of which is the new magnetic field. Hence,

HL = 3H

e3J + e−J

e3J + 3e−J

. (11.6.3)

In addition, Eq. (11.6.2) will give terms that are products of the spin variables on adjacentblocks, the coefficient of these gives the new coupling constant, and

JL = 2J

e3J + e−J

e3J + 3e−J

2

. (11.6.4)

At the fixed point, JL = J and HL = H . From Eq. (11.6.3) we see that H = 0 at the fixed point;i.e., the ferromagnetic critical point occurs at zero external magnetic field, as expected. Equation(11.6.4) shows that there are actually two fixed points: one with J = 0 and the other with

J =1

4ln(1 + 2√2) = 0. 335614. . . . (11.6.5)

The fixed point with J = 0 has no interactions and will not be considered further (actually, itdescribes the infinite temperature limit of the theory, and will yield mean-field behavior if ana-lyzed carefully). The other fixed point describes the finite-temperature critical point.

The linearized RG transformation, cf. Eq. (11.5.13), turns out to be diagonal with eigenval-ues

λ H = 3

e3J + e−J

e3J + 3e−J

(11.6.6)

and

λ J =2(e4J + 1)(e8J + 16Je4J + 4e4J + 3)

(e4J + 3)3(11.6.7)

10More generally, one must carry out what is known as a cumulant expansion; namely,

< eλ A >= exp

∞

j=1Σ λ j

j!<< A j >>

,

where << A j >> is known as a cumulant average. It is expressed in terms of the usual moment averages,< An >, by expanding both sides of the equation in a series in λ and comparing terms. To first order, itturns out that << A >>=< A >, which gives Eq. (11.6.2). It is also easy to show that<< A2 >>=< (A− < A >)2 > is just the variance. For more information on cumulants and moments, see,e.g., R. Kubo, Proc. Phys. Soc. Japan 17, 1100, 1962.

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which gives the numerical values shown in Table 11.6. The exponents were obtained from Eq.

(11.5.17), remembering that L = √3.

Table 11.6. Some Numerical Values

Quantity J* = ln(1 + 2√2) / 4 Exact‡

λ H 3/√2 = 2. 1213 2.80yH 1.3691 15/16

λ J 1.6235 √3 = 1. 73yJ 0.8822 0.5

α - 0.2671 0β 0.7152 0.125δ 2.1699 15

‡The 2d Ising model has an exact solution, first given by Onsager.

The agreement with the exact results isn’t great, the error mainly coming form λ H , and ismainly due to our use of first order perturbation theory for the spin potential of mean force, W .While it is conceptually simple to carry out the perturbation calculation to higher order, at theexpense of a lot of algebra, to do so requires that we consider some of the extra terms in theHamiltonian. The higher order calculation automatically generates interactions beyond nearestneighbors and beyond pairwise additive ones, and some of these must be considered if goodagreement is to be obtained. When this is done, the correct exponents and thermodynamic func-tions are found (to about 5% accuracy). An example of some of the better results is shown inFig. 11.10.

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Fig. 11.10. Results of higher order numerical RG calculation for the 2dlattice of Nieuhuis and Nauenberg11. The solid curve is the exact CH ,the dashed curve is the free energy, and the dot-dashed curve is theenergy, all from Onsager’s exact solution of the model. The points arethe numerical results. K is J in the text.

11.7. Concluding Remarks

We hav e accomplished the goals set out in Sec. 11.1. We now see how phase transitionsarise, why universal behavior is expected, and perhaps, most important, have provided a frame-work in which to calculate the critical exponents. Several important issues have not been dealtwith.

11B. Nienhuis and M. Nauenberg, Phys. Rev. B11, 4152, 1975.

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For example, other than the fact that the simple mean field approaches didn’t giv e the cor-rect experimental answer, we still don’t really understand why they failed, especially since manyof the qualitative features of a phase transition were described correctly. There is a consistent,albeit complicated, way in which to do perturbation theory on a partition function, which givesmean field theory as the leading order result. If we were to examine the next corrections, wewould see that they become large as the critical point is reached, thereby signaling the break-down of mean field theory. What is more interesting, is that the dimensionality of space plays akey role in this breakdown; in fact, perturbation theory doesn’t fail for spatial dimensions greaterthan four.

The dependence on the dimensionality of space plays a key role in Wilson’s work on criti-cal phenomena; in short, he’s (along with some key collaborators) have shown how to useε ≡ 4 − d as a small parameter in order to consistently move between mean field and non-meanfield behavior.

These, and other, issues require better tools for performing the perturbative analysis andfor considering very general models with complicated sets of interactions. This will not be pur-sued here, but the interested reader should have a look at the books by Ma3 or by Amit12 for adiscussion of the more advanced topics.

12D.J Amit, Field Theory, the Renormalization Group, and Critical Phenomena (McGraw-Hill, Inc.,1978).

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Liouville´s Equation -83- Chemistry 593

12. Liouville´s Equation

12.1. Classical Mechanics

The microscopic dynamics of classical systems is governed by Newton’s equations ofmotion. These can be cast in several different forms, in particular in the form known as Hamil-ton’s equations, i.e.,

pi = −∂H

∂qi

and qi =∂H

∂ pi

, i = 1, . . . , N , (12.1.1)

where pi and qi are known as the generalized momentum and coordinates, respectively, N is thenumber of degrees of freedom, and H is the Hamiltonian. For example, for a point moleculedescription of a gas or liquid, in the absence of magnetic fields, we write

H =N

iΣ p2

i

2mi

+ U(r1, . . . , rN ) (12.1.2)

and it is a simple matter to show that Newton’s equations of motion result from using Eq.(12.1.2) in (12.1.1) (the qi’s are associated with the various coordinates of the positions of thedifferent particles). Note that in 3D there are 3N degrees of freedom.

We are often interested in how functions of the coordinates and momentum change in time,either due to the motion of the particles or due to some explicit time dependence in the function(e.g., like we saw in the scattered electric field in light scattering where the time dependence ofthe scattered field arose from the product of the incident field and the dielectric fluctuations). Ifthe quantity of interest is denoted as A, it follows that

dA

dt=

∂A

∂t+

jΣ p j

∂A

∂ p j

+ q j

∂A

∂q j

=∂A

∂t+

jΣ −

∂H

∂q j

∂A

∂ p j

+∂H

∂ p j

∂A

∂q j

(12.1.3a)

≡∂A

∂t+ A, H ≡

∂A

∂t+ iLA, (12.1.3b)

where Eq. (12.1.3a follows from Hamilton’s equations (cf. Eq. (12.1.1)), where

A, B ≡j

Σ ∂A

∂q j

∂B

∂ p j

−∂A

∂ p j

∂B

∂q j

(12.1.4)

is known as a Poisson bracket, and the operator L is called the classical Liouville operator.

Equation (12.1.3b) strongly resembles the Heisenberg form for the time dependence of aquantum mechanical operator, the main change being the replacement of the Poisson bracket bya commutator, specifically, by letting

, →[ , ]

ih−, (12.1.5)

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Chemistry 593 -84- Liouville´s Equation

as will be discussed below.

12.2. Liouville’s Theorem in Classical Mechanics

As a classical system evolves dynamically the momenta and coordinates change in somecomplicated manner, and in doing so, the 6N dimensional phase vector, X(t), traces out a trajec-tory in phase space. The governing equations (whether in the form of Newton’s or Hamilton’sequations) are ordinary differential equations, and hence X(t) depends on the initial values of themomenta and coordinates; i.e., X(t) → X(X(0), t).

There is a remarkable theorem that can be proven about classical systems, referred to asLiouville’s Theorem, which states that under the action of the equations of motion, the phase vol-ume is conserved. In order to prove this, note that in any integration over phase space (e.g., likethe ones we do in ensemble averages) if we were to change variables from the momenta andcoordinates at time t to those at time 0, we would need the Jacobian for the transformation

J(t) ≡

∂ p1(t)

∂ p1(0).

.∂qN (t)

∂ p1(0)

. . .

. . .

. . .

. . .

∂ p1(t)

∂qN (0).

.∂qN (t)

∂qN (0)

. (12.2.1)

Equations of motion for the various partial derivatives that appear in Eq. (12.2.1) can be obtainedfrom Hamilton’s equations, cf. Eq. (12.1.1), i.e.,

d

dt

∂ pi(t)

∂xk(0)

=∂ pi(t)

∂xk(0)= −

∂∂xk(0)

∂H

∂qi(t)

= −N

j=1Σ ∂2 H

∂ p j(t)∂qi(t)

∂ p j(t)

∂xk(0)+

∂2 H

∂q j(t)∂qi(t)

∂q j(t)

∂xk(0)

(12.2.2a)

and

d

dt

∂qi(t)

∂xk(0)

=∂qi(t)

∂xk(0)=

∂∂xk(0)

∂H

∂ pi(t)

=N

j=1Σ ∂2 H

∂ p j(t)∂ pi(t)

∂ p j(t)

∂xk(0)+

∂2 H

∂q j(t)∂ pi(t)

∂q j(t)

∂xk(0),

(12.2.2b)

where xk(0) ≡ qk(0) or pk(0), k = 1, . . . , N . Note that the system of differential equations ofthe form given in Eqs.(12.2.2a) and (12.2.2b) is the same for each choice of xk(0); they only dif-fer in the initial conditions! Moreover, while these equations are messy, they hav e one importantfeature, namely they are linear in the partial derivatives we are interested in; specifically, thoseappearing in the Jacobian. Hence, the derivatives appearing in the Jacobian satisfy a linear sys-tem of ordinary differential equations (with time dependent coefficients). Moreover, viewed as asystem of ordinary differential equations, it is easy to see that the Jacobian we need is just theWronskian for the system of differential equations, usually denoted as W .

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We can now finish the proof of Liouville’s theorem by noting that Wronskians of systemsof linear equations satisfy Abel’s formula; i.e., if

y(t) = M(t)y(t) (12.2.3)

then

W (t) = W (0) exp∫

t

0dt Tr(M(t))

. (12.2.4)

For our problem, it’s easy to see that Tr(M(t)) = 0 and thus,

J(t) = J(0) = 1.

Hence, the size of the volume element in phase space doesn’t change as the particles move! Thisis known as Liouville’s theorem.*

A key application of Liouville’s theorem to statistical mechanics follows by noting thatsince each point in phase space represents a unique trajectory of the system. Since different tra-jectories don’t cross, all the trajectories started inside some phase element will remain inside thatphase element as time progresses (albeit with the shape of the phase element changing in somecomplex manner). Since the volume of the phase element also doesn’t change, we see that theprobability density (the number of trajectories per unit volume) remains constant; i.e.,

df N (t, X(t))

dt= 0, (12.2.5)

or using Eq. (12.1.3),

∂ fN

∂t= − iLf N . (12.2.6)

This is known as Liouville’s equation and governs the evolution of the probability density.

For equilibrium systems we require that Lf N = 0, and it is easy to see that any function ofconstants of the motion (energy, total momentum, etc.) will work.

12.3. Quantum Mechanics

In quantum systems, you might think that everything can be determined once a solution toShroedinger’s equation,

ih−∂Ψ∂t

= HΨ, (12.3.1)

is known. This isn’t quite right, and to see why, consider what happens if we express the general

*The manipulations leading to this result are similar to those leading to Bendixon’s Neg ative Criterion,which is used to study the stability of two dimensional systems of ODE’s. See, e.g., H.K. Wilson, Ordi-

nary Differential Equations (Introductory and Intermediate Courses Using Matrix Methods), (Addison-Wesley Pub. Co., Reading MA, 1971), Sec. 9.4.

Winter, 2018


solution to Eq. (12.3.1), for a time-independent Hamiltonian, as a superposition of energy eigen-functions, namely as

Ψ(t) =j

Σ e−iE j t/h−u jc j , (12.3.2)

where u j is the energy eigenfunction with energy E j and c j is some (complex) number. Anyobservable, O, is obtained through its expectation value, < O >, as

< O > = < Ψ(t)|O|Ψ(t) > =j, j′Σ c*

jc j′eiω j, j′t < j|O| j′ > , (12.3.3)

where ω j, j′ ≡ (E j − E j′)/h− is the transition frequency between energy states j and j′.

The preceding discussion is appropriate for cases where there is a definite wav e-function,i.e., for a pure state. Howev er, imagine what happens if we conduct a series of independent mea-surements and average the results. We would thus measure

< O >(t) =j, j′Σ ρ j′, je

iω j, j′t < j|O| j′ > , (12.3.4)

where the bar denotes the average over the different experiments and where

ρ j, j′ ≡ c jc*j′ (12.3.5)

is called the density matrix; it is determined by the statistical properties of the ensemble of mea-surements, and is said to describe a mixed state. For example, in equilibrium statistical mechan-ics ρ j, j′ = δ j, j′e

−β E j /Q, in the canonical ensemble.

The density matrix has several general properties that follow from its definition, cf. Eq.(12.3.5). For example, it is Hermitian (ρ j, j′ = ρ*

j′, j), normalized (Tr(ρ) = 1) and the diagonalelements are positive, and give the probabilities of being in state j. In addition, it is positive defi-nite; that is for any set of complex numbers, α j,

j, j′Σ α j ρ j, j′α

*j′ ≥ 0

and from this one can prove the Schwarz inequality

|ρ j, j′|2 ≤ ρ j, j ρ j′, j′,

where the equality holds for j = j′ and in general for pure states.†

In the Heisenberg representation, the operators, and not the wav e function, carry the timedependence of the system. Specifically, in the energy representation,

< j|OH (t)| j′ > ≡ eiω j, j′t < j|O(t)| j′ > , (12.3.6)

† The inequality can also be used to show that the density matrix of a pure state cannot evolve into one cor-responding to a mixed state under the action of Shroedinger’s equation. This raises some interesting ques-tions for the meaning of Ergodicity in quantum systems.

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where we have again allowed for the possibility that the operators explicitly depend on time. Bytaking time derivatives, it’s easy to show that the Heisenberg operators satisfy

dOH (t)

dt=

∂OH (t)

∂t+

[OH (t), H]

ih−≡

∂OH (t)

∂t+ iLOH (t), (12.3.7)

which is analogous to Eq. (12.1.3b), and results in the correspondence given in Eq. (12.1.5).

Once the time dependence of the operator is known, the ensemble averaged expectationvalue is easily obtained from Eqs. (12.3.4), namely,

< O >(t) =j, j′Σ ρ j′, j < j|OH (t)| j′ > = Tr(ρOH (t)), (12.3.8)

where Tr is the quantum mechanical trace. Alternately, we can associate the explicit time depen-dence with the density matrix; i.e.,

ρ j′, j(t) ≡ ρ j′, jeiω j, j′t (12.3.9)

in the energy representation. Now we hav e

< O >(t) = Tr(ρ(t)O). (12.3.10)

Hence, according to Eqs. (12.3.8) and (12.3.10) we can either propagate the operator and averageover some initial probabilistic distribution, or we can propagate the distribution and simply aver-age over the possible states of the operator; the same result is obtained.

Finally, if we take the time derivative of Eq. (12.3.9) we see that

∂ρ(t)

∂t= −

[ρ(t), H]

ih−. (12.3.11)

This is known as the quantum Liouville equation, and reduces to the classical one if the corre-spondence given in Eq. (12.1.5) is made.

One final note; the proofs used discussion have assumed that the Hamiltonian is time inde-pendent, and hence, that one has definite energy eigenvalues etc. Nonetheless, it is straightfor-ward to extend the discussion to the more general case by rewriting Eq. (12.3.2) as

Ψ(t) =j

Σ c j(t)u j , (12.3.12)

where now the u j are some complete set of functions and where

ih−dc j(t)

dt=

j′Σ < j|H(t)| j′ > c j′(t). (12.3.13)

The Heisenberg forms of the equations of motion remain unchanged as do the definition of thedensity matrix. Specifically,

ρ j, j′(t) ≡ c j(t)c*j′(t) (12.3.14)

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which still gives Eq. (12.3.11) for the time derivative.

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Nonequilibrium Systems -89- Chemistry 593

13. Nonequilibrium Systems

13.1. Linear Response Theory

As we have seen in the preceding handout, the fundamental microscopic equation govern-ing the behavior of the distribution function or density matrix in classical or quantum nonequilib-rium statistical mechanics is the Liouville equation, i.e.,

∂ f (t)

∂t= − iL(t) f (t), (13.1.1)

where f (t) is the distribution function or density matrix and the Liouville operator is defined as

iL(t)A ≡

A, H(t), classical[A, H(t)]

ih−, quantum.

(13.1.2)

and where . . . , . . . or [. . . , . . . ] denote a Poisson bracket or commutator, respectively. Forwhat follows, we write

H(t) = H0 + H1(t) (13.1.3a)

and

L(t) = L0 + L1(t), (13.1.3b)

where H0 governs the behavior of our system at equilibrium (and thus is not explicitly timedependent), while H1(t) is responsible for taking the system out of equilibrium, and as such, isexplicitly time dependent. In addition, we make two assumptions:

a) We start from an equilibrium canonical system in the distant past; i.e., H1(t) → 0 andf (t) → feq ≡ e−β H0 /Q as t → −∞.

and

b) H1(t) is small enough such that perturbation theory is valid.

In order to use this, we note the solution to Liouville’s equation also satisfies

f (t) = feq − ∫t

−∞ds e−iL0(t−s)iL1(s) f (s), (13.1.4)

as can be verified by differentiating both sides of Eq. (13.1.4) in time and showing that that theLiouville equation, cf. Eq. (13.1.1), is recovered (HINT: remember that iL0 feq = 0).

The advantage of rewriting Liouville’s equation this way is that the integral term is linearin the perturbation, and hence, can be thought of (formally) as being small. If we iterate Eq.

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Chemistry 593 -90- Nonequilibrium Systems

(13.1.4), e.g.,

f (t) = feq − ∫t

−∞ds e−iL0(t−s)iL1(s)[ feq − ∫

s

−∞ds1 e−iL0(s−s1)iL1(s1) f (s1)], (13.1.5)

a series in the strength of the perturbation is obtained. In particular, stopping at linear order gives

f (t) ≈ feq − ∫t

−∞ds e−iL0(t−s)iL1(s) feq, (13.1.6)

which is known as linear response theory, and was first introduced by Kubo1. Note that there aresome serious problems with using perturbation theory for many-body problems, as has been dis-cussed at length by van Kampen2. Nonetheless, it is generally believed that we can use Eq.(13.1.6) to generate expansions for the average behavior of few-body functions (e.g., like thenumber, energy or momentum densities).

At this point, some of the details of the discussion depend on whether we use the quantumor classical description. Since the former is slightly more complicated (not to mention correct)we’ll use it. We proceed by using Eq. (13.1.6) to compute the average deviation from its equilib-rium average value of some observable B. As we’ve discussed before,

< B(t) > = Tr((Bf (t))), (13.1.7)

and hence, by multiplying Eq. (13.1.6) on the left by B and taking the trace it follows that

b(t) ≡ < B(t) > − < B >eq = − ∫t

−∞ds

ih−Tr

Be−iH0(t−s)/h−[ feq, H1(s)]eiH0(t−s)/h−

, (13.1.8)

where we have expressed the quantum mechanical Liouville propagator, e−iL0t in terms of theusual quantum mechanical propagators, e−iH0t/h−. By noting that the trace is invariant to cyclicpermutations, i.e.,

Tr(ABC) = Tr(BCA) = Tr(CAB), (13.1.9)

and that in the Heisenberg representation

B(t) = eiL0t B = eiH0t/h− Be−iH0t/h−, (13.1.10)

it follows that

b(t) = − ∫t

−∞ds

ih−Tr

B(t − s)[ feq, H1(s)]

= ∫t

−∞ds

< [B(t − s), H1(s)] >eq

ih−. (13.1.11)

In order to proceed, we have to specify the form of the perturbation. A very general one is:

H1(X N , t) ≡ −αΣ ∫ dr Aα ((r, X N ))Fα (t) ≡ − A * F(t), (13.1.12)

1R. Kubo, J. Phys. Soc. Japan 12, 570 (1957).2N. G. van Kampen, Physica Novegica 5, 279 (1971).

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where * denotes a dot product and integration over space and where the Aα (r, t) are quantitieslike the microscopic number, charge, dipole, etc., densities; they depend implicitly on timethrough the motion of the particles or evolution of the wav e function. The Fα (r, t) correspond tothings like external electric, magnetic, gravitational fields and do not depend on the implicitdynamics or motion of the system.

When Eq. (13.1.12b) is used in Eq. (13.1.11), we find that

b(t) = ∫t

−∞ds

i

h−< [B(t − s), A] >eq *F(s). (13.1.13)

Notice that both the dynamics and average refer to the equilibrium system.

Before discussing the implications of this result, note that the classical limit can beobtained either by repeating the steps following Eq. (13.1.6) using classical mechanics andreplacing the trace by an integration over phase space, or by using the standard correspondence

[. . . , . . . ]

ih−→ . . . , . . . .

The same result is obtained, and thus,

b(t) = − ∫t

−∞ds < B(t − s), A >eq *F(s). (13.1.14a)

= − β ∫t

−∞ds < B(t − s), H0A >eq *F(s) (13.1.14b)

= − β ∫t

−∞ds < B(t − s)A >eq *F(s), (13.1.14c)

where Eq. (13.1.14b) is obtained by using the explicit form of the Poisson bracket and integrat-ing by parts, while Eq. (13.1.14c) is obtained when the rate of change of a mechanical variable isexpressed in Poisson bracket form (cf. the last handout). Remember that Eqs. (13.1.14) holdonly in the classical limit; in general, Eq. (13.1.13) must be used.

The quantum mechanical result can be made to look like the classical one using the follow-ing trick. By again using the invariance of the trace to cyclic permutations we can rewrite Eq.(13.1.13) as

b(t) = ∫t

−∞ds

i

h− QTr

e−β H0

eiH0(t−s)/h− Be−iH0(t−s)/h− − eiH0(t−s)/h−+β H0 Be−iH0(t−s)/h−−β H0

A

* F(s).

= ∫t

−∞ds

1

ih−< [B(t − s − iβ h−) − B(t − s)]A >eq *F(s)

= − β ∫t

−∞ds < BK (t − s)A >eq *F(s), (13.1.15)

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where BK is called a Kubo transform and is defined as

BK (t) ≡ ∫1

0dλ B(t − iβ h− λ). (13.1.16)

Hence, we’ve been able to express the nonequilibrium behavior of a system in terms ofexpressions containing equilibrium time correlation functions of various types. There are addi-tional properties of time correlation functions that are sometimes useful, and some of these willbe discussed below. Finally, note that there are other ways in which to arrive at the linearresponse expressions and to extend them beyond linear order3.

13.2. Response Functions, Susceptibilities and The Fluctuation Dissipation Theorem

The basic form of our result is independent of whether we use classical or quantummechanics, namely,

b(t) =t

∫−∞ds χ B,A(t − s) * F(s). (13.2.1)

Up to factors of i and 1/2, χ is referred to as a response function4 and plays the same role as aGreen’s function. Note that it is causal, i.e., the response of the system at time t only depends onthe F’s for earlier times.

It is instructive to look at the Fourier transform of Eq. (13.2.1) as this will give theresponse of the system to a monochromatic perturbation (only to linear order of course). By let-ting

f (ω ) ≡ ∫∞−∞

dt e−iω t f (t) (13.2.2)

and noting that Eq. (13.2.1) is a convolution integral, it follows that

b(ω ) = χ B,A(ω ) * F(ω ), (13.2.3)

where

χ B,A(ω ) ≡ ∫∞0

dt e−iω t χ B,A(t). (13.2.4)

Equation (13.2.3) is easily identified as an generalization of AC susceptibility. In fact, if we sub-ject the neutral system to a uniform external electric field, E(t), and use Eq. (13.2.4) to computethe response of the total electric polarization, i.e., B ≡ P and H1 ≡ − P ⋅ E(t), we find the usualelectric susceptibility,5

3See, e.g., D. Ronis and I. Oppenheim, Nonlinear Response Theory for Inhomogeneous Systems in Various

Ensembles, Physica (Utrecht) 86A, 475 (1977) or D. Ronis, Statistical Mechanics of Systems Nonlinearly

Displaced from Equilibrium I, Physica (Utrecht) 99A, 403 (1979).4See, e.g., D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (W.A.Benjamin, Inc., Reading Massachusetts, 1975).5Actually this isn’t quite right, since the definition of the susceptibility is in terms of the local electric field,not the external one. For a giv en sample geometry it is not to hard to correct for this, and in any case, is

Winter, 2018


χ el(ω ) = ∫∞0

dt e−iω t i

h−< [P(t), P] > ,

where henceforth, we drop the "eq" notation on all equilibrium averages. Similar expressions areeasily obtained for the magnetic susceptibility, conductivity, etc..

One important and often easily measurable quantity is the power absorbed by the system,P. From mechanics, it follows that

P =∂H

∂t=

∂H1(t)

∂t= − A *

∂F(t)

∂t, (13.2.5)

which when ensemble averaged, using Eq. (13.1.15), gives

< P(t) > ∼∂F(t)

∂t* β ∫

t

−∞ds < AK (t − s)A > *F(s). (13.2.6)

By Fourier transforming in time we find that

< P(ω ) > ∼β ∫dω1

2πi(ω − ω1)F(ω − ω1) *

− < AK (t = 0) A > +iω1 < AK (ω1)A >

* F(ω1),

(13.2.7)

where remember that the Fourier transform of the correlation function is half-sided (cf. Eq.(13.2.4)) and where we have integrated by parts. For ω = 0, it is easy to show that the contribu-tion to the integrand from the equal time correlation function is an odd function of ω1 and henceits integral vanishes; we thus find that the the total energy dissipated by the system, Ptotal , is

Ptotal∼β ∫dω1

2πω 2

1 F*(ω1)* < AK (ω1)A > *F(ω1), (13.2.8)

where we have used the fact that f (−ω ) = f*(ω ) for real functions f (t).

Under quite general conditions, one can prove that the right hand side of Eq. (13.2.8) isnonnegative,6 and hence, net power is always absorbed in any experiment. Also note, that underfairly mild assumptions about the response function’s Fourier transform, the contribution to thedissipation will vanish as the frequency of the perturbation tends to zero. This is an example ofthe fact all finite rate processes are accompanied with dissipation (loss, friction) and are thus irre-versible. Finally, note that equilibrium scattering experiments (like light, neutron, x-ray) mea-sure equilibrium time correlation functions of the type that appear in Eq. (13.2.8); thus, there is adirect experimental connection between the equilibrium fluctuations and the nonequilibriumresponse or dissipation. Equation (13.2.8) is known as the fluctuation dissipation theorem.There are some extra details which will be omitted here, however, see, e.g., Ref. 4.

not very important for weakly polarizable systems.6The proof is based on Bochner’s Theorem, cf. F. Riesz and B. Sz-Nagy, Functional Analysis, (FredrickUngar Pub. Co., New York, 1972).

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13.3. Relaxation Experiments

In one important class of experiments, a system is allowed to equilibrate in some sort offield, which is shut off at t = 0, and the relaxation back to the original equilibrium is followed,e.g., as depicted in Fig. 13.1.

Fig. 13.1. A schematic representation of a relaxation experi-ment. The system is prepared before t = 0 (here exxageratedin the sence that the preparation stage should be flatter). Att = 0 the constraints supporting the steady state are removedand the system relaxes to the new equilibrium. Examples thatare oscillatory or monotonic are known. Note that this kind ofmeasurement is sometimes referred to as a free inductiondecay (FID).

As long as the initial deviation from equilibrium is small, we can use linear response theory. Tomodel the equilibration process, we write,

F(t) ≡

eε t F , for t ≤ 0

0, for t > 0,(13.3.1)

where we assume that ε → 0 + in order to eliminate any transient effects associated with prepar-ing the system. When Eq. (13.3.1) is used in Eq. (13.1.15) and the result integrated by parts it iseasy to show that

b(t) =

β < BK (0)A > *Feε t , for t ≤ 0

β < BK (t)A > *F , for t > 0,(13.3.2)

where terms O(ε ) hav e been dropped, and where the ˆ is still used to denote the deviation fromthe equilibrium average value.

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The behavior for t ≤ 0 is exactly what is expected if the system comes to some sort of adia-batic- or local-equilibrium; i.e, if it is governed by an equilibrium distribution with HamiltonianH = H0 − A * F . For example, in this case, using classical statistical mechanics for a canonicalensemble, it follows that

< B >LE ≡ ∫ dX N B(X N )e−β (H0−A*F)

∫ dX N e−β (H0−A*F)=

< Beβ A*F >

< eβ A*F >

∼ < B > +β < B A > *F + O(F2). (13.3.3)

Up to this point, we have discussed the response theory in terms of the F’s, consideringthem as real external fields. However, in the relaxation experiment, as we have just seen, theironly role is to provide an initial condition. Consider the trivial case, where there is only oneexternal field F coupling to the density of some trace component. For long times and lengthscales, we expect diffusive behavior, i.e., the concentration of our tracer should obey

∂c(r, t)

∂t= D∇2c(r, t), (13.3.4)

where the parameter D is called the diffusion constant. One way to solve this equation is byFourier transforming in space, thereby converting Eq. (13.3.4) into a simple first order ODE,with solution

c(k, t) = e−Dk2t c(k, 0). (13.3.5)

The initial condition can be obtained from F using Eq. (13.3.3) or vise versa. However if wecompare Eqs. (13.3.2) and (13.3.5) for long times, we expect that

< C(k, t)C(−k) > ∼e−Dk2t (13.3.6)

which decays very slowly for small wav e-numbers.

Computing such correlations, e.g., by a direct molecular dynamics simulation of the requi-site correlation function is going to be extremely difficult, given the need to go to long times.This situation will occur whenever slowly evolving quantities are encountered (e.g., as is alwaysthe case with the densities of conserved quantities). To make matters worse, even though weexpect the macroscopic equations like the diffusion equation [Eq. (13.3.4)] to hold in many situa-tions, the form of the solutions depends not just on initial conditions but also on the boundaryconditions of the problem; hence, not only would we have to perform a very long time simula-tion, but we’d hav e to simulate a large enough system to stand a chance of capturing the physicsof the boundary conditions. The way around these problems is to focus on the equations ofmotion and not on their solutions.

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13.4. Microscopic and Macroscopic Equations Of Motion

13.4.1. Dynamical Variables

The problem encountered in the last section arose because we looked at something thatev olved on a slow time scale. In the example of the tracer density, this happened because of aconservation principle, namely, conservation of mass. More generally, slow dynamics will arisefor the density of any conserved quantity, but may also arise in systems with long range order, orwhere the dynamics is slow because of the nature of the molecules or phases (e.g., in polymers orglasses). Here we will focus mainly on the former.

For density a conserved quantity, the microscopic equations of motion must have the formof a continuity equation; i.e.,

∂A(r, t)

∂t= − ∇ ⋅ J(r, t), (13.4.1)

where J(r, t) is called the flux. The reason Eq. (13.4.1) must hold is that it guarantees that thetotal rate of change of a inside some volume is caused by transport at the surface of the volume(as is easily shown, using the divergence theorem). Another way to see this is to Fourier trans-form Eq. (13.4.1) in space, which gives:

∂ A(k, t)

∂t= ik ⋅ J(k, t), (13.4.2)

again showing that things evolve slowly when long wav elength phenomena are considered.

Microscopic expressions for the basic conserved quantities, the so-called hydrodynamicvariables, (density, momentum and energy) in a classical system comprised of point masses inter-acting by pairwise additive forces are given in the following table:

Table 13.I. Microscopic Densities of Conserved Quantities

Name Symbol A(r, t) A(k, t)

Number Nj

Σ δ (r − r j)j

Σ eik⋅r j

Momentum Pj

Σ δ (r − r j)p jj

Σ p jeik⋅r j

Energy Ej

Σ δ (r − r j)

p2j

2m+

1

2 j′≠ jΣ u j, j′

j

Σ eik⋅r j

p2j

2m+

1

2 j′≠ jΣ u j, j′

With these definitions it is easy to show that

∂N (k, t)

∂t= ik ⋅ P(k, t)/m, (13.4.3a)

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∂P(k, t)

∂t= ik ⋅ ↔τ (k, t), (13.4.3b)

and

∂E(k, t)

∂t= ik ⋅ JE (k, t), (13.4.3c)

where

↔τ (k, t) ≡j

Σ eik⋅r j

p jp j

m+

1

2 j′≠ jΣ r j, j′F j, j′

1 − e−ik⋅r j, j′

ik ⋅ r j, j′

(13.4.4a)

and

JE (k, t) ≡j

Σ eik⋅r j

p j

m

p2j

m+

1

2 j′≠ jΣ u j, j′

+1

2 j′≠ jΣ r j, j′F j, j′ ⋅

p j

m

1 − e−ik⋅r j, j′

ik ⋅ r j, j′

(13.4.4b)

are referred to as the stress tensor and energy current, respectively. Note that for a one-compo-nent system the number (mass) flux is simply proportional to the total momentum density; this isnot the case for mixtures.

13.4.2. Some Properties of Correlation Functions

For what follows, it is more convenient to consider spatial Fourier transforms of the equa-tions. First however, some general properties of time correlation functions and their transformsmust be considered. Consider

C AB(r − r′, t − t′) ≡ < A(r, t)B(r′, t′) > , (13.4.5)

where C AB can be written as a function of r − r′ for translationally invariant systems (such as liq-uids and gases) and t − t′ since the correlation function is defined in an equilibrium ensemble. Ifwe consider any of the correlation functions encountered above, it is clear that they will all havethis form and hence, one has to deal with convolution integrals in space associated with the *operation (and also in time for the general external perturbation). The spatial convolution disap-pears if we Fourier transform convolutions in space, i.e.,

∫ dr1 C AB(r − r1, t − t′)F(r1) → C AB(k, t − t′)F(k).

There are several ways to represent the Fourier transform of C AB. For example, if we simplyFourier transform Eq. (13.4.5), it is easy to show that

< A(k, t)B(k′, t′) > = δk+k′,0VC AB(k, t − t′), (13.4.6a)

where V is the volume of the system and δk,k′ is a Kronecker-δ . Rearranging Eq. (13.4.6a) fork′ = −k gives

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C AB(k, t − t′) =< A(k, t)B(−k, t′) >

V. (13.4.6b)

Another useful property is the limiting behavior as k → 0. In general, we can write

C AB(r − r′, t − t′) = < A(r) >< B(r′) > + < [A(r, t)− < A(r) >][B(r′, t′)− < B(r′)] >(13.4.7)

where we expect that the second term is short ranged (at least with respect to the size of the sys-tem) and where the first term is constant for uniform systems. When Fourier transformed, thefirst terms generate δ -functions at k = 0, and hence, do not appear in the limit. Thus,

k→0lim C AB(k, t − t′) =

< AT (t)BT (t′) >

V, (13.4.8)

where the T subscript denotes integration over the total volume of the system and the ˆ impliesthe deviation from the equilibrium average value. In addition, there is some subtlety about thechoice of ensemble used to calculate the average appearing in Eq. (13.4.8); the correct ensembleis the grand-canonical ensemble.7 One important consequence of this result is that the limit isindependent of time when either A or B is the density of a conserved quantity. In this case, theresulting average is often just one of the variances encountered in the study of fluctuations inequilibrium statistical mechanics, and as such, can be related to thermodynamic quantities. Forexample, it is easy to show that, for A = [N , E, P]T

< AT AT

T >

V=

∂N /V

∂β µ

β ,V

−

∂N /V

∂ββ µ,V

0

−

∂N /V

∂ββ µ,V

−

∂E/V

∂ββ µ,V

0

0

0

mρkBT↔1

(13.4.9a)

=

kBT ρ2κ

kBT 2 ρ

κ H

VT− α

0

kBT 2 ρ

κ H

VT− α

kBT 2

CV

V+

Tακ

− pα −

κ H

VT

0

0

0

mkBT ρ↔1

, (13.4.9b)

where ρ ≡ < N > /V , κ is the isothermal compressibility, α is the volume expansion coefficient,H is the enthalpy, and CV is the constant volume heat capacity.

7See, e.g., http://ronispc.chem.mcgill.ca/ronis/chem646/structure_factor.1.html

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13.4.3. Equations of Motion

We now hav e the results we need in order to get the equations of motion. Henceforth, clas-sical expressions will be used (use the Kubo transform otherwise). We start by rewriting theresult for the t ≥ 0 response of a system, cf. Eq. (13.3.2), in Fourier representation, i.e.,

b(k, t) = β < B(k, t)A(−k) > ⋅F(k)/V , (13.4.10)

where the ˜ used to denote a Fourier transform will be dropped when confusion won’t arise. Aswas discussed above, Eq. (13.4.10) gives the solution to the Eqs. of motion in terms of the F ,which, cf. Eq. (13.3.3), essentially are determined by initial conditions. This is not what wewant.

We assume that a closed nonequilibrium description (e.g., like the diffusion equation) ispossible in terms of some small set of nonequilibrium state variables, like, for example, the den-sities of the conserved quantities. Minimally, we assume that the applied fields must be generalenough to adjust the initial averages of these state variables to arbitrary values, and hence, A

must include these, whether or not real external fields can be found that do this.8 This was ini-tially recognized by Kubo1 who calls the F’s fictitious forces for this reason.

By using Eq. (13.4.10) for B = A it follows that

β F(k) = V < A(k, t)A(−k) >−1 ⋅a(k, t), (13.4.11)

which when used in Eq. (13.4.10) gives

b(k, t) = < B(k, t)A(−k) > ⋅ < A(k, t)A(−k) >−1 ⋅a(k, t). (13.4.12)

In particular, with B = A we find

a(k, t) = M(k, t) ⋅ a(k, t), (13.4.13)

where

M(k, t) ≡ < A(k, t)A(−k) > ⋅ < A(k, t)A(−k) >−1 . (13.4.14)

Thus, a closed system of ODE’s for the evolution of the state variables has been obtained. Ofcourse, you shouldn’t be fooled, ultimately, Eq. (13.4.13), depends on the assumed relaxationexperiment, cf. Eq. (13.3.1), and this is why the coefficients of the equations of motion explicitlydepend on time.

If a linear phenomenology in terms of the A is valid for long times, M(t) will becometime-independent. In order to see why this might be the case, we rewrite Eq. (13.4.14) as

M(k, t) = M(k, 0) + ∫t

0ds M(k, s)

8For example, no real external field will produce a local-equilibrium system with a nonuniform tempera-ture.

Winter, 2018


= < A(k)A(−k) >< A(k)A(−k) >−1 − ∫t

0ds < A

‡(k, s) A

‡(−k) >< A(k, s)A(−k) >−1 ,

(13.4.15)

where

A‡(k, t) ≡ A(k, t) − M(k, t) A(k, t) (13.4.16)

is referred to as the dissipative part of A(k, t). In obtaining these last results, we had to be care-ful when computing the derivatives of matrices (which in general don’t commute) and have usedthe Hermiticity of the Liouville operator, which implies that

< A(t)B(t′) > = − < A(t)B(t′) > . (13.4.17)

This is sometimes called the dot-switching property of time correlation functions, and can also beobtained by noting that the time correlation functions are functions of t − t′.

Equation (13.4.16) is extremely interesting; basically it gives the part of the microscopic

rate of change, not described by the macroscopic linear equations of motion, cf. Eq. (13.4.13)

and (13.4.14), as applied to the microscopic variables. Thus, A‡

would be identically zero if themacroscopic equations held microscopically e.g., as is the case for the number density in a onecomponent system [cf. Eq. (13.4.3a)]. Of course, in general this is not the case; however, to theextent that linear equations of motion correctly describe the dynamics for long time macroscopicphenomena we expect that correlations involving the dissipative parts will decay on a much

shorter time scale.9 Therefore for macroscopic times, we expect that < A‡(k, s) A

‡(−k) >→ 0,

and hence,

M(k, t) = < A(k)A(−k) >< A(k)A(−k) >−1 − ∫∞0

ds < A‡(k, s) A

‡(−k) >< A(k, s)A(−k) >−1 .

(13.4.18)

Moreover, we expect that the A‡

time correlation function to decay on microscopic time scales,thereby becoming amenable to direct computer simulation. Note that if you’ve guessed thewrong phenomenology, i.e., by choosing the wrong variables A, then the correlations may notdecay as expected, however, the converse in not true; just because the integral in Eq. (13.4.18)converges is no proof that the correct nonequilibrium state variables have been found.

The two terms on the right hand side of Eq. (13.4.18) are referred to as being re versible

and dissipative, respectively. It can be shown that the former are related to local-equilibrium cor-rections to local-equilibrium average rates of change like those encountered in Eq. (13.3.3). Thelatter are solely responsible for the increase of entropy associated with the second law of

9In fact, this is also not correct. In general the macroscopic equations of motion are not linear and thisresults in small, slowly decaying long-time tails in the dissipative correlation functions. In two dimen-sions, this problem leads to logarithmically divergent integrals and in general makes

t→∞lim M(k, t) non-ana-

lytic in k.

Winter, 2018


thermodynamics.

For the hydrodynamic variables, it is easy to show that the reversible terms are odd func-tions of k, while the dissipative terms are proportional to k2, cf. Eqs. (13.4.2) or (13.4.3). Thus,for small k (long wav elength phenomena) we can rewrite Eq. (13.4.18) as

M(k, t)∼ik⋅ < JT AT > −kk: ∫

∞0

ds < J‡T (s)J‡

T >

< AT AT >−1 +O(k3), (13.4.19)

where

J‡T (s) ≡ J

T(s)− < JT AT >< AT AT >−1 AT (13.4.20)

and is referred to as a dissipative current. Note that it will vanish whenever the microscopic fluxis one of the variables in A

The inverse of the variance matrix can be explicitly calculated from Eqs. (13.4.9a) or(13.4.9b); however, it is simpler to note that

V < AT AT

T >−1 =

∂β µ

∂N /V

E,V

∂(−β )

∂N /V

E,V

0

∂β µ

∂E/V

N ,V

∂(−β )

∂E/V

N ,V

0

0

0

↔1/(mρkBT )

, (13.4.21)

which allows us to introduce

β Φ(k, t) ≡ V < AT AT

T >−1 ⋅a(k, t) ≡ [δ (β µ)(k, t), −δ β (k, t), β v(k, t)]T , (13.4.22)

where we have defined (here to linear order) the Fourier transforms of the deviations of tempera-ture and chemical potentials from equilibrium through their equilibrium Taylor expansions interms of number and energy densities. In addition, the local fluid velocity , v, is defined in termsof the momentum density as v = p/(mρ), as expected. That nonequilibrium versions of chemicalpotential and temperature have arisen should not come as a surprise. At equilibrium, thermody-namics requires that the gradients of these vanish, but says nothing about the number and energydensities; hence, the rates should be proportional to the gradients of these quantities, and thesearise naturally.

The integrals appearing in Eq. (13.4.18) are called Onsager coefficients, i.e.,

Li, j ≡ ∫∞0

ds< J

‡i,T (s)J

‡j,T >

V, (13.4.23)

and the integral expression is called a Green-Kubo relation. There is a remarkable symmetryprinciple, called reciprocal-relations, proved by Onsager (for which he won the Chemistry NobelPrize in 1968). It is trivially obtained using Eq. (13.4.23). From the explicit expressions for thefluxes (cf., Eqs. (13.4.4a) or (13.4.4b)) it follows that they hav e definite parity under inversion of

Winter, 2018


the momenta; i.e.,

Ji → ε iJi , for p1 → −p1, . . . , (13.4.24)

where ε i = ± 1. In classical mechanics, for systems not subject to velocity dependent fields (e.g.,magnetic fields), reversing all the momenta will cause the trajectory to run backwards in time, aproperty referred to as time-reversal symmetry. If we were to do this in the average in Eq.(13.4.23) we find that

Li, j = ε iε j ∫∞0

ds

V< J

‡i,T (−s)J

‡j,T >= ε iε j L j,i (13.4.25)

where the last equality follows when the stationarity of the equilibrium correlation function isused. It is particularly useful in multi-component systems where there are many diffusionOnsager coefficients and for the phenomena known as the Soret and Dufore effects (where a tem-perature gradient excites a diffusion current and visa versa).

13.4.4. The Hydrodynamic Equations

We end our discussion by considering the explicit forms of the equations of motion for thehydrodynamic equations.

13.4.4.1. Number Density

The number density in a one component system is particularly trivial. As was pointed out

above, J‡N ,T = 0 and it is easy to show that the reversible terms give:

∂n(k, t)

∂t= ik ⋅ p(k, t)/m, (13.4.26)

as expected from the exact form, cf. Eq. (13.4.3a).

13.4.4.2. Energy Density

The energy density has both reversible and irreversible terms. However, by noting that theenergy current, cf. Eq. (13.4.4b), is an odd function of the momentum, it follows that< JT ,E AT ,α > is nonzero only for α = P; moreover, by using the so-called virial form for theequilibrium pressure, ph,

<↔τ T > =

↔1 phV (13.4.27)

and explicitly carrying out the average over momenta, it follows that

< JT ,E PT >

NmkBT=

h

mρ, (13.4.28)

where h = e + ph is the equilibrium enthalpy per unit volume.

The Onsager coefficients associated with the energy current, cf. Eq. (13.4.23), in principle

give hav e nonzero couplings to energy or momentum (remember that J‡N = 0, since the

Winter, 2018


momentum density is one of the nonequilibrium state variables). The terms coupling to momen-tum involve a third-rank object, cf. Eq. (13.4.23), and it is easy to show that averages of all oddorder functions of momenta and coordinates vanish in systems with inversion symmetry, i.e.,under the transformation X N → −XN . This only leaves the dissipative couplings to the energy,and here too symmetry implies that for isotropic systems that the x − x, y − y and z − z elementsare all equal, while the others vanish. This together with the result for the reversible terms, Eq.(13.4.28), Eq. (13.4.19), and our definition of the nonequilibrium temperature, cf. Eq. (13.4.22),gives

∂e(k, t)

∂t= hik ⋅ v(k, t) − λ k2δ T (k, t), (13.4.29)

where the thermal conductivity, λ is given as

λ ≡ ∫∞0

ds< J

‡T ,E (s) ⋅ J

‡T ,E >

VkBT 2, (13.4.30)

δ (−β )∼δ T /(kBT 2), and where

J‡T ,E (t) ≡ JT ,E (t) −

h

ρPT (t), (13.4.31)

cf. Eq. (13.4.20).

13.4.4.3. Momentum Density

The most complicated of the hydrodynamic equations are those associated with themomentum. Beginning with the reversible terms, note that since the stress tensor, cf. Eq.

(13.4.4a), is an even function of the momenta, it follows that <↔τ T PT >= 0. The remaining

reversible terms, cf. Eq. (13.4.19), involve

<↔τ T NT >

V=

∂ <

↔τ T > /V

∂β µ

β ,V

=↔1

∂ ph

∂β µ

β ,V

(13.4.32a)

and

<↔τ T ET >

V=

∂ <

↔τ T > /V

∂(−β )

β µ,V

=↔1

∂ ph

∂(−β )

β µ,V

, (13.4.32b)

where in each of Eqs. (13.4.32a) and (13.4.32b), the first equalities follow using the usual manip-ulations encountered in the theory of fluctuations and where the second equalities follow fromEq. (13.4.27). When these results are combined with Eqs. (13.4.21) and (13.4.22), we see that thereversible terms are just the Taylor expansion of a local-equilibrium pressure:

δ ph(r, t)∼

∂ ph

∂β µ

β

δ (β µ)(r, t) +

∂ ph

∂ββ µ

δ β (r, t) (13.4.33)

(actually of its spatial Fourier transform). The key point is that the derivatives are all equilibrium

Winter, 2018


properties that are easily expressed in terms of common thermodynamic quantities. In addition,it is trivial to reexpress the pressure deviation in terms of other variables (like n or e).

The couplings to between stress and energy current in the dissipative current vanish forsystems with inversion symmetry (as expected from Onsager’s reciprocal relations). The remain-ing Onsager coefficients involve couplings between the dissipative stress tensor and itself, afourth rank object:

β Li, j;k,lP,P ≡ ∫

∞0

ds< τ i, j

T

‡(s)τ k,l

T

‡>

VkBT, (13.4.34)

where

↔τ‡

T (t) = ↔τ T (t) −↔1

∂ ph

∂n

e

NT −

∂ ph

∂e

n

ET

. (13.4.35)

Fortunately, in isotropic systems, symmetry drastically cuts down on the number of independentparameters. First, note that τ i, j is symmetric in i and j, for systems interacting through centralforces, cf. Eq. (13.4.4a). Second, remember that in an isotropic system our choice of coordinate

axis shouldn’t change any results. Hence, the form of Li, j;k,lP,P must be invariant to any kind of

rotation of the coordinate axis. The only way to do this is to express Li, j;k,lP,P as a linear combina-

tion of various identity matrices (or Kronecker-δ ’s). Thus,

β Li, j;k,lP,P = ζ δ i, jδ k,l + η(δ i,kδ j,l + δ i,lδ j,k −

2

3

δ i, jδ k,l), (13.4.36)

where ζ and η are known as the bulk and kinematic viscosities, respectively. From Eqs.(13.4.34) and (13.4.36) it follows that

η = ∫∞0

ds< τ x,y

T (s)τ x,yT >

VkBT(13.4.37a)

and

ζ = ∫∞0

ds< Tr(

↔τ

‡

T )(s)Tr(↔τ

‡

T ) >

9VkBT. (13.4.37b)

Note that the ‡ can be omitted from the off-diagonal elements of↔τ T because the subtractions are

diagonal, cf. Eq. (13.4.35). This makes the kinematic viscosity particularly amenable to numeri-cal computation. In addition, the kinematic viscosity is usually much more important than thebulk viscosity in fluid flows.

When Eqs. (13.4.33) and (13.4.36) are used in Eq. (13.4.19) and the various contractions(sums over the Cartesian indices) are performed we get

∂p(k, t)

∂t= ikδ ph(k, t) − k2ηv(k, t) − (ζ +

1

3

η)kk ⋅ v(k, t) (13.4.38)

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which is just the space Fourier transform of the linearized Navier-Stokes equations of hydrody-namics.

13.5. Concluding Remarks

We conclude by summarizing all our results for the hydrodynamic equations in the coordi-nate representation:

∂n(r, t)

∂t= − ∇ ⋅ p(r, t)/m, (13.5.1)

∂e(r, t)

∂t= − h∇ ⋅ v(r, t) + λ∇2T (r, t), (13.5.2)

and

∂p(r, t)

∂t= − ∇δ ph(r, t) + η∇2v(r, t) + (ζ +

1

3

η)∇[∇ ⋅ v(r, t)]. (13.5.3)

Of course, these equations have been around for quite some time. What is new here is the con-nection to the microscopic world through the Green-Kubo relations, in particular, Eqs. (13.4.30),(13.4.37a) and (13.4.38b). It is an interesting exercise to extend the calculation to a multi-com-ponent system, and in particular to one which is infinitely dilute in a tracer, thereby obtaining amicroscopic expression for the self-diffusion coefficient.

Note that the methods presented here are not restricted to deriving equations of motion; byreturning to Eq. (13.4.12) and choosing the B appropriately many other things can be examined.Some examples that have been looked at include, fluctuations in nonequilibrium systems (e.g., asmeasured in scattering experiments) and reduced nonequilibrium distributions.

Winter, 2018

Chemistry 593 -106- Langevin Equations

14. Langevin Equations

14.1. The Langevin Equation

Our discussion of response theory has led to the forms of the macroscopic equations ofmotion and the parameters that appear in them in terms microscopic time correlation functions,the so-called Green-Kubo relations. Historically, this is not how this field developed.

Consider a brightly lit room. If you examine The illuminated regions, you will see thescattering from floating dust particles that undergo a seemingly random motion. One compo-nent of this is called Brownian Motion,1 although there are other sources of this motion, e.g.,convection. If we imagine the dust particles as small non-interacting spheres, we expect them toobey an equation of motion that minimally includes the friction the particle feels as it moves; i.e.,

mdv(t)

dt= −ξ v(t), (14.1)

where m is the particle mass and ξ is called the friction coefficient. By using hydrodynamics,Stokes showed that

ξ = 6πη R, (14.2)

where R is the particle radius and η is the dynamic viscosity of the host media. You expect thatthis is valid for large enough particles; it turns out, with some minor changes, that it seems tohold at the microscopic scales as well.2

Equation (14.1) is easily solved and gives,

v(t) = e−t/τ v(0), (14.3)

where τ ≡ m/ξ . There is no randomness at all! To fix this, we modify Eq. (14.1) by adding arandom term to the right hand side of Eq. (14.3) which becomes

mdv(t)

dt= −ξ v(t) + F‡(t), (14.4)

where F‡(t) is called the random force and is analogous to what appears in the Green-Kuboexpressions we obtained earlier. This equation is an example of a stochastic equation and iscalled the Langevin equation. Since F‡(t) accounts for processes not covered by the linear (aver-age) frictional term we expect

⟨F‡(t)⟩ = 0 and ⟨F‡(t)F‡(t′)⟩ = 2χ↔1δ (t − t′), (14.5)

1A. Einstein, Investigations on the Theory of the Brownian Movement, (Dover Publications, New Your,N.Y, 1956).2See, e.g., B.J. Bern and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and

Physics (John Wiley & Sons, Inc., New York, NY, 1976), ch. 7.

Winter, 2018

Langevin Equations -107- Chemistry 593

where δ (x) is the Dirac δ -function. It could be modeled by any short ranged function of t − t′.Note that it is commonly assumed that F‡(t) is Gaussian distributed, something we don’t need forthe present discussion.

The general solution to Eq. (14.4) is

v(t) = e−ξ t/mv(0) + ∫t

0ds e−ξ (t−s)/mF‡(s)/m. (14.6)

By averaging over the bath degrees of freedom3 we see that

⟨v(t)⟩ = e−ξ t/mv(0) (14.7a)

and

⟨v(t)v(t′)⟩ = e−ξ (t+t′)/mv(0)v(0) + ∫t

0ds ∫

t′

0ds′ e−ξ (t+t′−s−s′)/m 2χ

m2

↔1δ (s − s′)

= e−ξ (t+t′)/mv(0)v(0) + ∫min(t,t′)

0ds e−ξ (t+t′−2s)/m 2χ

m2

↔1

= e−ξ (t+t′)/mv(0)v(0) −

χξ m

↔1

+ e−ξ |t−t′|/m χξ m

↔1. (14.7b)

By taking the limit t, t′ → ∞, keeping t = t′ it follows that

⟨vv⟩ =kBT

m=

χξ m

,

where the first equality is obtained by assuming an equilibrium Maxwell Boltzmann velocity dis-tribution and where the second follows from Eq. (14.7b). The final result can be rearranged toshow that

χ = kBTξ , (14.8)

which is an example of what is known as an Einstein-Nyquist relation.4 Equation (14.8) alsoshows that fluctuations are related to dissipation; you can’t hav e one without the other and expectthe system to equilibrate (although it will come to a steady-state). Note that the transient termsin Eq. (14.7b) vanish if the initial conditions are averaged using the equilibrium Maxwell-Boltz-mann velocity distribution.

Finally, once the initial transients decay, we see that

3This is analogous to what we did when we introduced the reduced correlation functions and/or potentialsof mean force.4These relations and their generalizations, are sometime erroneously referred to as fluctuation-dissipationrelations.

Winter, 2018

Chemistry 593 -108- Langevin Equations

⟨v(t)v(t′)⟩ = e−ξ |t−t′|/m kBT↔1

m. (14.9)

What is the connection to the response theory discussion we had in the previous section?If we let A ≡ P, where P is the Brownian particle’s momentum, the equation of motion duringthe relaxation experiment becomes

d⟨P(t)⟩NE

dt= ⟨F(t)P⟩ ⋅ ⟨P(t)P⟩−1 ⋅ ⟨P(t)⟩NE =

⟨F(t) ⋅ P⟩⟨P(t) ⋅ P⟩

⟨P(t)⟩NE , (14.10)

where F(t) is the force the bath particles exert on the large one. Note that the second equalityfollows when isotropy is used to show that the equilibrium time correlation functions are diago-nal. By repeating the manipulations in the preceding chapter, it follows that

d⟨P(t)⟩NE

dt= −ζ ⟨P(t)⟩NE , (14.11)

where

ζ ≡ ∫t

0ds

⟨F‡(s) ⋅ F‡⟩⟨P(s) ⋅ P⟩

, (14.12)

with

F‡(s) ≡ F(s) −⟨F(s) ⋅ P⟩⟨P(s) ⋅ P⟩

P(s). (14.13)

If we assume that the random force correlation functions decay quickly compared to the Brown-ian particle’s momentum it follows that

⟨F(s) ⋅ P⟩⟨P(s) ⋅ P⟩

≈⟨F ⋅ P⟩⟨P ⋅ P⟩

=0

3mkBT= 0, (14.14)

which implies, cf. Eq. (14.13), that F‡(t) ≈ F(t) and allows us to rewrite the friction coefficient as

ζ ≈ ∫∞0

ds⟨F(s) ⋅ F⟩3mkBT

. (14.15)

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Projection Operators -109- Chemistry 593

15. Projection Operators

15.1. Inner Products in Quantum Statistical Mechanics

For any two quantum mechanical operators, A and B, let their inner product be

(A, B) ≡ < BA†K > = < BK A† > , (15.1)

where < . . . > denotes an equilibrium canonical average, the K subscript denotes a Kubo trans-form, i.e.,

AK ≡ ∫1

0dλ A(−iβ h− λ) = ∫

1

0dλ eβ λ H Ae−β λ H , (15.2)

and where the second equality follows by cyclically permuting the operators under the trace. Inthe energy representation, with eigenstates states labeled by n with energies En,

(A, B) =n,mΣ e−β En

QBn,m ∫

1

0dλ eβ λ En A*

n,me−β λ Em (15.3)

=n,mΣ e−β En

QBn,m A*

n,m

eβ (En−Em) − 1

β (En − Em)

. (15.4)

Note that (ex − 1) /x is a monotonically increasing positive function. From Eq. (15.4) it followsthat:

1. (A, B) = (B, A)*,

2. For any complex scalar α , (A, α B) = α (A, B),

3. (A, B + C) = (A, B) + (A, C),

and

4. ||A||2 ≡ (A, A) is real and positive unless A = 0 (i.e., all An,m vanish).

These are just the required properties of an inner product; hence, we can view the space of opera-tors in quantum mechanics as an inner-product space, and with some assumptions about com-pleteness (convergence properties of so-called Cauchy sequences), as a Hilbert space.

15.2. The Projection Operator Identity

Consider two operators, O1 and O2, neither of which has any explicit time dependence,and may or may not commute with each other. It is easy to show that

e(O1+O2)t = eO1t + ∫t

0ds eO1(t−s)O2e(O1+O2)s, (15.5a)

which is similar to the identity we used to convert Liouville’s equation into an integral equation

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Chemistry 593 -110- Projection Operators

suitable for generating the perturbative expansion resulting in linear response theory.* Here ourgoal is somewhat different, and we let O1 ≡ iL and O2 ≡ −PiL, where L is the Liouville operator(classical or quantum) and P a projection operator, specifically an operator that is Hermitian andindepotent , i.e., P2 = P. With these choices of O1 and O2 are used in Eq. (15.5a) and the resultrearranged, it follows that

eiLt = ei(1−P)Lt + ∫t

0ds eiL(t−s)iPLei(1−P)Ls, (15.5b)

which, when used in the equation of motion for a mechanical variable, A, giv es:

A(t) = eiLt iLA = eiLt(P + 1 − P)iLA = eiLt PiLA + A‡(t) + ∫

t

0ds eiL(t−s)PiL A

‡(s), (15.6)

where

A‡(t) ≡ e(1−P)iLt(1 − P)iLA, (15.7)

and is often referred to as the random force. Note that P A‡(t) = 0 for all t; i.e., it is orthogonal

to the subspace projected out by P.

In order to proceed we have to specify the form of the projection operator. There are manyways to do this, and here, we will use the so-called Mori projection operator1,2; specifically,

PB ≡ A(A, A)−1(A, B) = < BK A† > ⋅ < AK A† >−1 ⋅A, (15.8)

where A denotes a set of mechanical variables that will serve as a basis for the subspace of inter-est and where the second equality is obtained when Eq. (15.1) is used for the inner product. Notethat (A, A)−1 is the matrix inverse of the matrix whose elements are (Aα , Aβ ).

With this definition of the projection operator, we can rewrite Eq. (15.6b) as

A(t) = iΩA(t) − ∫t

0ds K(t − s)A(s) + A

‡(t), (15.9)

where

*Note that Eq. (15.5a) can be proved by Laplace transforming both sides of the equality or by showing theboth sides obey the same equation of motion and are trivially equal at t = 0.1H, Mori, Prog. Theort. Phys. (Japan) 34, 423 (1965). For a more general discussion, see, D. Forster,Hydrodyanmic Fluctuations, Broken Symmetry, and Correlation Functions, (W.A. Benjamin, Inc., Read-ing, MA, 1975).2Another choice, due to Zwanzig, is to partially trace some of the degrees of freedom. For example, if weuse a separable basis for the wav e functions, e.g., spin and translational wav efunctions, we can define aprojection operator by

PB = TransTr (Be−β H )

TransTr (e−β H )

,

where the trace is only over the translational degrees of freedom.

Winter, 2018

Projection Operators -111- Chemistry 593

iΩ ≡ [(A, A)−1(A, iL A)]T = < AK A† >< AK A† >−1 (15.10)

and

K(t) ≡ − [(A, A)−1(A, iL A‡(t))]T = < A

‡K (t) A

†>< AK A† >−1 (15.11)

is the memory function. In obtaining the last equality, the hermiticity of the Liouville operator

(or the dot-switching property of equilibrium averages) was used. Since, by construction, A‡(t)

is orthogonal to A for all times, it follows that < A‡(t)A

‡K >= 0; this allows us to rewrite the defi-

nition of the memory function as

K(t) = < A‡K (t) A

‡†

>< AK A† >−1 (15.12)

Equation (15.9) is referred to as a generalized Langevin equation, and Eq. (15.12) is a gen-eralization of the famous Einstein-Nyquist relations; they are formally exact. Note however:

1. The time dependence of memory functions is not directly computable in that the dynamics isnot governed by Newtonian or Heisenberg dynamics; i.e., the propagator that appears isexp[(1 − P)iLt] and not exp(iLt). Nonetheless, there are special cases (e.g., long-wav elengthdynamics in hydrodynamics, or the large mass Brownian particle) where the differences go tozero.

2. The statistics of the "random" force terms is in general unknown.

3. The appearance of "0" as the lower limit of integration in our memory equation. The systemdoesn’t know how we choose to define the origin of time. Similarly, the projection operatorexplicitly depends on the choice of the equilibrium canonical distribution, why this is alwaysrelevant in a nonequilibrium situation (at least initially) is not clear.

4. The results will depend on the choice of the basis used in the projection operator, i.e., on theA’s. Clearly, if we’re interested in long-time dynamics, we expect the basis to include allslow dynamical variables. These include:

a) Densities of conserved quantities.

b) A constant (all this does is to replace A by A ev erywhere).

c) Broken symmetry variables.

d) Any other slow variables, e.g., the momentum of a heavy particle in a bath of light ones.

e) Nonlinear combinations of any of the above. This last property is a serious technicalcomplication, however, in three dimensions, away from critical points, the fluctuationsare small, and the couplings to the nonlinear combinations are weak.

In any event, for a properly chosen basis, we expect A‡(t) to evolve on the fast time scale, and as

such, K(t) should decay on a microscopic time scale, τ micro, or at least on one that is much fasterthan that characterizing the evolution of the A’s; hence, when there is this good separation oftime scales, Eq. (15.9) can be approximately written as:

Winter, 2018

Chemistry 593 -112- Projection Operators

A(t) = iΩA(t) − ∫∞0

ds K(s)A(t) + A‡(t), (15.13)

which is simply a multivariate Langevin equation.

More generally, we can solve Eq. (15.9) by using Laplace transforms. This gives:

A(z) = z − iΩ + K(z)

−1

A(0) + ˜A

‡(z)

, (15.14)

where the tilde denotes a Laplace transform, z is the transform variable, and

K(z) ≡ ∫∞0

dt e−zt < A‡K (t) A

‡†

>< AK A† >−1 (15.15)

is a frequency dependent, generalized friction constant matrix; it becomes identical to the oneappearing Eq.(15.13) for zτ micro << 1.

One interesting application of Eq. (15.14) is in the computation of time correlation func-

tions of the type that appear in linear response theories, i.e., Ci, j(t) ≡ < Ai,K (t)A†j >. From Eq.

(15.14), again using the orthogonality of A‡

and A, it follows that

C(z) = z − iΩ + K(z)

−1

C(0), (15.16)

which is an example of the Onsager regression hypothesis. Alternately, we can use this as a wayto calculate the memory function by rearranging Eq. (15.16) as

K(z) = C(0)C−1

(z) − z + iΩ. (15.17)

Note that several of the objections raised above do not apply to the application of the projectionoperator method to equilibrium time correlation functions; in particular, the distribution is knownand t denotes the time separation, not an absolute time.

Winter, 2018

Inelastic Light Scattering -113- Chemistry 593

16. Inelastic Light Scattering

16.1. The Interaction of Radiation and Fluctuations

i iωk

n

nf

P.M.T.

to compter

Laser

Sample

θ

i

ωk f

f

Optical Filter:

Heterodyne Mixing

Fabrey−Perot, or

Fig. 16.1. Schematic of a Light Scattering Experiment

The simplest scattering experiment is cartooned in the preceding figure. Basically, the out-put from a monochromatic light source (a laser) is passed through a polarizing filter that choosesan initial polarization for the incident light (ni) and then passes through the sample. Most of thelight simply passes through the sample unaffected, however instantaneous inhomogeneities willscatter some of the light into different directions and these are picked up by a detector set atsome scattering angle θ . In the simplest experiment, the detector is a photo-multiplier tube(PMT) sitting behind a polarizing filter that selects some final polarization. More sophisticatedsetups may include various kinds of optical filtering devices as shown. As we shall see, thedetector may have to be modified in order to probe inelastic scattering processes, but that will bediscussed elsewhere. The interaction of the incident light with the medium in the sample, is gov-erned by Maxwell’s equations. Here the discussion of light-scattering presented in class will besummarized. The starting point is Maxwell’s Equations for a system with no free charges or cur-rents; i.e.,

∇ ⋅ D = 0 (Coulomb′s Law), (16.1a)

∇ ⋅ B = 0 (no magnetic monopoles), (16.1b)

curl E = −1

c0

∂B

∂t(Faraday′s Law), (16.1c)

and

curl H =1

c0

∂D

∂t(Amper e′s Law with Maxwell′s displacement current), (16.1d)

Winter, 2018

Chemistry 593 -114- Inelastic Light Scattering

where so-called Gaussian units have been used, c0 = 3 × 108m/s is the speed of light in vacuum,E and B are the electric field and magnetic induction or magnetic flux density, respectively, D isthe electric displacement field, and H is the magnetic field. See the appendix for definitions andsome properties of div, grad, and curl. The displacement fields account for the polarization ofmatter due to the presence of electric or magnetic fields. In addition to Maxwell’s equations, wealso need a constitutive relation linking the displacement fields to the real electric and magneticfields. The simplest form for these relations are

D(r, t) = ↔ε (r, t) ⋅ E(r, t) (16.2a)

and

B(r, t) = ↔µ(r, t) ⋅ H(r, t) (16.2b)

where↔ε (r, t) and

↔µ(r, t) are the electric and magnetic permeabilities, respectively (the former is

also referred to as the dielectric tensor). For what follows, we shall ignore all magnetic effectsand set the magnetic permeability to unity. Note that Eqs. (16.2a) and (16.2b) are not the mostcomplicated expressions that could be assumed. For example, they assume that the response isinstantaneous and local, neither of which is exactly true.

Since the molecules in any sample at finite temperature move, there will be fluctuations inthe dielectric properties and it is these fluctuations that are responsible for the scattered light inthe system. To see how this comes about, we write

↔ε (r, t) = ε0

↔1 + δ ↔ε (r, t), (16.3)

where ⟨↔ε (r, t)⟩ = ε0

↔1 and where we have assumed that the medium is isotropic and homogeneous

on average.

In order to proceed, we take the curl of Faraday’s law, Eq. (16.1c) and eliminate curl B byusing Ampere’s law, Eq. (16.1d). This gives

1

c20

∂2D(r, t)

∂t2= −curl curl E(r, t) (16.4)

= ∇2E(r, t) − ∇divE(r, t), (16.5)

where the last equality follows by using the identity curl curl A = ∇divA − ∇2 A

Since we expect that fluctuations are small, we now dev elop a perturbative expansion inδ ↔ε (r, t). To leading order, (i.e., we set δ ↔ε (r, t) = 0), divE(0)(r, t) = 0, cf. Eq. (16.1a), and Eq.(16.5) becomes:

1

c2

∂2

∂t2− ∇2

E(0)(r, t) = 0, (16.6)

where c ≡ c0/√ ε0 is the speed of light in the material. Equation (16.6) is a wav e equation andwe’ll look for a wav e-like solution, specifically we try

Winter, 2018


E(0)(r, t) = E0ni exp[i(ω i t − ki ⋅ r)]. (16.7)

It is easy to see that this ansatz satisfies Eq. (16.6) as long as

ω i = kic, (16.8)

which is the usual dispersion relation for light. In addition, we assumed that ∇ ⋅ E(0)(r, t) = 0,which implies that

ki ⋅ ni = 0, (16.9)

that is, the polarization and propagation directions of the light are orthogonal. The magneticfield now follows by using Eq. (16.7) in Ampere’s law, Eq. (16.1c). Some simple calculus showsthat

B(0)(r, t) =c0(ki × ni)

ω i

E0 exp[i(ω i t − ki ⋅ r)], (16.10)

which shows that the magnetic field is transverse to both the propagation direction and to theelectric field. It is for this reason, that light is sometimes referred to as being transverse.

Thus, we see that in the absence of fluctuations, there are monochromatic plane-wav e solu-tions to Maxwell’s equations. The first order corrections due to fluctuations in the dielectric prop-erties of the medium follow by writing D(r, t) = D(0)(r, t) + D(1)(r, t)+. . . , etc., in Eq. (16.5). Therelation between D(1) and E(1) must be handled carefully. From Eqs. (16.2a) and (16.3) it fol-lows that

D(1)(r, t) = ε0E(1)(r, t) + δ ↔ε (r, t) ⋅ E(0)(r, t). (16.11)

By solving Eq. (16.11) for E(1)(r, t) and using the result in Eq. (16.5) we find that

1

c2

∂2

∂t2− ∇2

D(1)(r, t) = curl curl [δ ↔ε (r, t) ⋅ E(0)(r, t)], (16.12)

where we have again used the identity for curl curl, this time in reverse. Given the form of theright hand side of Eq. (16.12) and Eq. (16.1a), it is useful to introduce the Hertz vector, Π(r, t),by writing

D(1)(r, t) ≡ −curl curl Π(r, t), (16.13)

which when used in Eq. (16.12) shows that

1

c2

∂2

∂t2− ∇2

Π(r, t) = −δ ↔ε (r, t) ⋅ E(0)(r, t). (16.14)

The left hand side of this equation is the same as that which we found earlier, cf. Eq. (16.6).Now howev er, there is a nonzero right-hand side, which nonetheless depends only on the zerothorder solution and on the dielectric fluctuations; these will play the role of sources for the elec-tro-magnetic scattered wav es.

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We now turn to the solution of Eq. (16.14) for arbitrary sources. To do this we first con-sider a very special source, namely an instantaneous, localized disturbance at t = 0 and r = 0.For this case, we rewrite Eq. (16.13) as

1

c2

∂2

∂t2− ∇2

G(r, t) = −δ (r)δ (t), (16.15)

where δ (x) is the Dirac δ -function and G(r, t) is called the Green’s function. Equation (16.15)can be solved by Fourier transforms, specifically we write

G(r, t) ≡ ∫dkdω(2π )4

ei(ω t−k⋅r)g(k, ω ),

use this in Eq. (16.15) and then Fourier transform the resulting equation; this gives

g(k, ω ) =c2

ω 2 − (kc)2,

and thus,

G(r, t) = c2 ∫dkdω(2π )4

ei(ω t−k⋅r)

ω 2 − (kc)2.

The evaluation of the integrals in this last expression is complicated by the fact that the frequencyintegration includes the singularities at ω = ±kc. Moreover, it is not clear that G obeys the physi-cal requirement that the response follows the disturbance; i.e., that G(r, t) = 0 for t ≤ 0. Bothproblems are fixed by adjusting the path of integration over frequency. Specifically, the integra-tion path is distorted to loop around the singularities from below in the complex frequency plane,as shown in the Figure.

−kc kc

Complex Fequency Plane

Re(ω )

Im( ω )

Fig. 16.2. Frequency Integration Contour. If youknow some complex analysis (don’t worry too muchif you don’t1),

you can easily see that the frequency integral vanishes for t < 0 (the contour integral is closed

1A readable introduction to complex analysis can be found in R. Churchill, Complex Variables and Appli-

cations, (McGraw-Hill, New York).

Winter, 2018


below and the poles are outside the contour). For t > 0 the contour is closed above and Cauchy’stheorem shows that

∫dω2π

eiω t

ω 2 − (kc)2= −

sin(kct)

kc.

Alternately, we can obtain this result directly from Eq. (16.15) by Fourier transforming justin space. This gives

1

c2

∂2

∂t2+ k2

G(k, t) = −δ (t). (16.15a)

As was mentioned above, G(k, t) = 0 for t < 0, while for t > 0 (where δ (t) = 0) the last equationshows that

G(k, t) = A sin(kct) + B cos(kct),

where A and B are arbitrary constants. One is determined by requiring that the solution be con-tinuous at t = 0, and hence, B = 0. The remaining constant can be obtained by integrating Eq.(16.15a) in t between −ε and ε and subsequently letting ε → 0 +. This is known as the "jumpcondition" and gives

1

c2

∂G(k, t)

∂t

ε

−ε

= −1

which shows that A = −c/k, thereby giving the same result as the method based on complex anal-ysis.

Returning to the main discussion, we next invert the remaining spatial Fourier transform;i.e.,

G(r, t) = −c ∫dk

(2π )3e−ik⋅r sin(kct)

k.

Finally, we switch to polar coordinates with the z axis chosen to lie along r and perform the inte-grations over angles. This gives

G(r, t) = −c

2π 2r ∫∞0

dk sin(kr) sin(kct)

= −c

8π 2r ∫∞−∞

dk [cos((k(r − ct))) − cos((k(r + ct)))]

= −c

4π r[δ (r − ct) − δ (r + ct)], (16.16)

where we have used the Fourier representation of the δ -function:

Winter, 2018


∫∞−∞

dk

2πeikx = δ (x).

Only the first delta function can have a nonzero argument since we’ve assumed t > 0, and thus

G(r, t) = −c

4π rδ (r − ct). (16.17)

This means that the electro-magnetic response to an impulsive source is a spherically outgoingshell of radiation. The radius grows with the speed of light and falls off like r−1.

The next question is how to use Eq. (16.17) for our general problem, cf. Eq. (16.14). It iseasy to check, using Eq. (16.15) that

Π(r, t) = ∫ dr1dt1 G(r − r1, t − t1)δ ↔ε (r1, t1) ⋅ E(0)(r1, t1) (16.18)

indeed satisfies Eq. (16.14), and hence, by using Eq. (16.17) for G(r, t) we find that

Π(r, t) = − ∫ dr1

δ ↔ε (r1, tret) ⋅ E(0)(r1, tret)

4π |r − r1|, (16.19)

where the retarded time is given by

tret ≡ t −|r − r1|

c; (16.20)

i.e., the time in the past for which a disturbance at r1 reaches the point r at time t, assuming thatthe response propagates at the speed of light. For distances on the order of a meter, the amountof retardation is O(10−8 sec), and hence cannot be ignored for functions like E (0), which vary ona much faster time scale. On the other hand, for many, but certainly not all, interesting molecularprocesses, the variation of δ ↔ε (r, t) happens on a slow time scale. When this is the case, we canignore the effects of retardation in the dielectric fluctuations, and this is assumed for what fol-lows.*

In addition, we’ll examine our result in the far field; specifically, we assume that r >> r1,and we write |r − r1| ≈ r − r ⋅ r1+. . . , for the purpose of calculating the retardation times. Thus,the discussion of this and the preceding paragraph and the explicit form for E(0)(r, t), (cf. Eq.(16.7)), allow us to rewrite Eq. (16.19) as

Π(r, t) ≈ −E0ei(ω i t−kir)

4π r ∫ dr1 δ ↔ε (r1, t) ⋅ nieiq⋅r1 (16.21)

*For faster time-scale processes, this approximation cannot be made. Nonetheless, for a monochromaticincident plane wav e, the frequency Fourier spectrum of the Hertz field is easily obtained, and for opticalfiltering based detection schemes (see below) this is all that is required. By using Eq. (16.7) in (16.19) andperforming a time Fourier transform, it is straightforward to show that:

Π(r, ω ) = −E0 ∫ dr1

δ ↔ε (r1, ω − ω i) ⋅ nie−i[ω |r−r1|/c+ki ⋅r1]

4π |r − r1|∼ −

E0eik f r

4π rδ ↔ε (q, ω − ω i) ⋅ ni ,

where now k f ≡ ω r/c and where the last approximation follows in the far field, as above.

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where the momentum transfer q is given by

q ≡ k f − ki , (16.22)

where k f ≡ω i

cr = ki r is the wav evector of the scattered radiation. Note that q has exactly the

same form as was obtained in the simple model of elastic scattering.

To finish the calculation, we must compute the scattered fields, cf. Eq. (16.13). This task issimplified by noting that ki is large and that taking derivatives of 1/r results in functions thatdecay faster at long distances; hence, only derivatives of the exponential factor need be kept, andwe thus find that

D(1)(r, t) ≈ −E0ei(ω i t−kir)

4π rk f ×

k f × ∫ dr1 δ ↔ε (r1, t) ⋅ nie

iq⋅r1, (16.23)

where we have used the identity: curl(fΦ) = ∇Φ × f + Φcurl(f). This result further simplifies ifwe assume that the detected light passes through a polarizing filter that selects polarization n f

(where n f ⋅ k f = 0). In this case what gets through to the detector is:

n f ⋅ D(1)(r, t) ≈E0k2

f ei(ω i t−kir)

4π rδ ε f ,i(q, t), (16.24)

where

δ ε f ,i(q, t) ≡ ∫ dr1 n f ⋅ δ ↔ε (r1, t) ⋅ nieiq⋅r1 . (16.25)

It is interesting to compare our result with that for the electric field of an oscillating dipole in thefar field (in vacuum, i.e., ε0 = 1)2, namely

E = −k f × (k f × p)ei(ω t−k f r)

r,

where p is the initial value of the dipole moment. We’v e obtained essentially the same result,provided we let

p ≡δ ε (q, t) ⋅ ni E0

4π.

Now δ ↔ε (r, t) = 4π δ ↔χ (r, t), where δ ↔χ (r, t) is the fluctuation in the local electric polarizabilitytensor. Since

↔χ (r, t) ⋅ E(r, t) is the instantaneous dipole moment induced per unit volume, ouridentification is reasonable (and becomes exact as q → 0.

The scattered magnetic field can be obtained from Eq. (16.23) using Eq. (16.1c) as we didabove. The result is that

B(1)(r, t) =c0

ω i

k f × E(1)(r, t), (16.26)

2See, e.g., J. D. Jackson, Classical Electrodynamics, (John Wiley and Sons, New York).

Winter, 2018


where we have again assumed that the largest derivatives are those associated with the the factorexp[i(ω i t − kir)]. Thus the scattered light is also transverse to it’s propagation direction (r).

To recap, we’ve shown that the scattered field has the form of a spherically outgoing wav e,modulated by a time- and scattering-angle-dependent factor, the Fourier transform of the dielec-tric tensor fluctuations. Other that the time dependence of the modulation factor, this is com-pletely analogous to what we found heuristically when considering elastic scattering.

16.2. Detecting the Scattered Radiation

As we’ve just seen the scattered fields have the form of spherically outgoing wav es thatoscillate at a frequency near that of the incident laser light (remember that δ ε f ,i(q, t) depends ontime and hence introduces small frequency shifts into the spectrum of scattered light). Thismakes the direct detection of the radiation very difficult, in that electronic devices do not typi-cally respond on the 10−15sec time scale. Fortunately this doesn’t matter since, most detectorsmeasure the power in the scattered radiation and not the actual fields. The radiative flux (energyper unit time per unit area) is called the Poynting vector (cf. Ref. 2) and is defined as

S(r, t) =c

8πRe[E(r, t) × H*(r, t)] (16.27)

where the * denotes the complex conjugate (this comes about when we take the real parts of thescattered electric and magnetic fields calculated above, compute the real form of the Poyntingvector, and ignore terms that oscillate at roughly twice the optical frequency). By using theexplicit forms for the fields obtained in the last section, assuming that the scattered light ispassed through a polarizer, it follows that the power entering the detector is:

I (t) ≡ S(r, t) ⋅ r =ck4

f

16π 2r2|δ ε f ,i(q, t)|2 I0 (16.28)

where I0 ≡ c|E0|2/8π is the incident power per unit area.

Photo-multiplier tubes typically respond on microsecond time scales, times long comparedto those of most (but not all) molecular processes. This means that the PMT is defacto comput-ing a long time average of the scattered intensity, and hence, we can convert it to an ensembleav erage. (What ensemble do you think is most appropriate?). Thus

I (t)

I0

=ck4

f

16π 2r2⟨|δ ε f ,i(q, t)|2⟩. (16.29)

On first glance, the average in Eq. (16.29) is slightly different from those we’ve encounteredbefore, in that the quantity being averaged depends on time. However, since it is an equilibriumav erage, the actual time is immaterial, and thus, can be set to zero. This result is almost the sameas that which we obtained when considering elastic light scattering. The exact connection to ourprevious treatment can be established by noting that the dielectric constant many materials isstrongly dependent on density, weakly dependent on temperature and isotropic; hence, it is rea-sonable to assume that

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δ ↔ε (r, t) ≈

∂ε0

∂ρ

T

δ N (r, t)↔1,

where N (r, t) is the local instantaneous number density. By using this in Eq. (16.29) it followsthat

I (t)

I0

=ck4

f

16π 2r2

∂ε0

∂ρ

T

2

⟨N (q)N*(q)⟩δ f ,i , (16.30)

where

N (q) ≡j

Σ eiq⋅r j , (16.31)

and δ f ,i is called a Kronecker-δ , it vanishes unless f = i, and is unity for f = i. Thus we’veobtained essentially the same result as before, although now we know the proportionality con-stants. Of course, Eq. (16.30) is approximate, and in particular predicts that no depolarizationcan occur (i.e., the scattered light has the same polarization as the incident beam).

It is relatively easy to include, at least approximately, the depolarization arising fromasymmetric molecules. By noting that

D(r, t) = E(r, t) + 4π P(r, t),

where

P(r, t) ≡iΣ µi(t)δ (r − ri(t))

is the polarization density and µi(t) is the induced or permanent dipole moment on the i’th mole-cule. Note that both induced and permanent dipoles depend on the orientation of the molecule,degrees of freedom that must be included in the average.

Consider a system of molecules without permanent dipole moments. In this case,µi = ↔α i(t) ⋅ E(ri , t),

P(r, t) ≡iΣ δ (r − ri(t))

↔α i(t) ⋅ E(r, t),

and

D(r, t) ≡

↔1 + 4π

iΣ δ (r − ri(t))

↔α i(t)

⋅ E(r, t).

Thus, the dielectric tensor can be identified as

↔ε (r, t) =↔1 + 4π

iΣ δ (r − ri(t))

↔α i(t),

where fluctuations arise because the density and molecular orientations fluctuate (the latterchanging

↔α i(t)). Finally, note that these examples assume local (in both space and time) relations

Winter, 2018


between P(r, t) and the local electric field, E(r, t).

16.3. Probing the Time Dependence of the Fluctuations

You might suspect that there might be some way in which to study the time dependence inthe scattered radiation resulting from the fluctuations in the dielectric tensor in Eq. (16.24).Indeed there are several ways in which to do this.

16.3.1. Optical Filtering Methods

In this approach, an optical frequency filter (analogous to an electronic band pass filter) isplaced in the optical path before the PMT. The most commonly used device is the so-calledFabry-Perot interferometer. The interferometer is tunable and only passes some narrow band offrequencies (in the optical range). How does this change the intensity measured at the PMT?

In order to see what happens, let’s write the scattered intensity in a Fourier series in time;i.e.,

E (1)(q, t) = ∫∞−∞

dω2π

eiω t E(1)

(q, ω ), (16.32)

where

E(1)

(q, ω ) ≡ ∫∞−∞

dt e−iω t E (1)(q, t), (16.33)

as usual.† What the optical filter does is to weight the different frequencies differently. If wedenote this weighting by some function f (ω ), the electric field that gets to the detector is now,

E (1)(q, t) = ∫∞−∞

dω2π

eiω t f (ω )E(1)

(q, ω ). (16.34)

The detector is the same PMT as before and this measures the power, essentially the square ofthe scattered field; hence,

I (t)

I0

∝ ∫∞−∞

dω2π ∫

∞−∞

dω ′2π

ei(ω −ω ′)t f (ω ) f *(ω ′)⟨E(1)

(q, ω )E(1)*

(q, ω ′)⟩. (16.35)

What can be said about the average that appears in the last expression? By using the defi-nitions of the Fourier transforms, cf. Eq. (16.33), it follows that

⟨E(1)

(q, ω )E(1)*

(q, ω ′)⟩ = ∫∞−∞

dt ∫∞−∞

dt′ e−i(ω t−ω ′t′)⟨E (1)(q, t)E (1)*

(q, t′)⟩, (16.36)

where the average on the right hand side is called a time correlation function. It has several gen-eral properties, and in particular, for equilibrium averages, it only depends on time differences(this is still true if the exp(−iω i t) factors are kept). Thus, we can write

†Note that the scattered field contains two time dependent factors, eiω i t and δ ε f ,i(q, t), cf. Eq (16.24).When Fourier transformed in time, the former results in a frequency shift; i.e., ω → ω − ω i and it is thisshifted scale that is used in Raman spectroscopy.

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⟨E (1)(q, t)E (1)*

(q, t′)⟩ = ⟨E (1)(q, t − t′)E (1)*

(q, 0)⟩

in Eq. (16.36). We do this and also change integration variables to τ ≡ t − t′ and T ≡ (t + t′)/2(the Jacobian for the transformation is unity). Some simple algebra shows that

⟨E(1)

(q, ω )E(1)*

(q, ω ′)⟩ = ∫∞−∞

dTdτ e−i(ω −ω ′)T −i(ω +ω ′)τ /2⟨E (1)(q, τ )E (1)*

(q, 0)⟩,

= (2π )δ (ω − ω ′) ∫∞−∞

dt e−iω t⟨E (1)(q, t)E (1)*

(q, 0)⟩, (16.37)

where we’ve used the Fourier representation of the δ -function. When Eq. (16.37) is used back inEq. (16.35) we find that

I (t)

I0

∝ ∫∞−∞

dω2π

| f (ω )|2C(ω ), (16.38)

where

C(ω ) ≡ ∫∞−∞

dt e−iω t⟨E (1)(q, t)E (1)*

(q, 0)⟩ (16.39)

is the Fourier transform of the scattered electric field time correlation function.‡ Hence, by tuningthe interferometer, we can map out the Fourier transform of the scattered field correlations intime, and in turn, these are simply proportional to the fluctuations in the dielectric tensor.

In practice the frequency resolution of a good Fabry-Perot interferometer is O(10MHz),and this imposes a upper limit on the time scale of the phenomena that can be studied with thistechnique.

16.3.2. Optical Mixing Methods

16.3.2.1. Heterodyne Mixing

In order to probe longer time scale phenomena, the detection scheme must be changed.There are several ways this can be done, and here we’ll focus only on one of them, namely het-erodyne mixing. The experiment is cartooned below.

‡By using Eq. (16.24), it follows that

C(ω )∝ ∫∞

−∞dt e−i(ω −ω i)t⟨δ ε f ,i(q, t)δ ε *

f ,i(q, 0)⟩.

Hence, like in Raman spectroscopy, the dielectric fluctuations introduce a time-dependent modulation tothe incident beam which is resolvable as long as the Fabry-Perot interferometer has a resolving power,∆ω < 1/τ , where τ is the characteristic timescale of the dielectric fluctuations. Slower processes don’tintroduce enough of a frequency shift to be separated by the interferometer.

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i iωk

n

nf

P.M.T.

to autocorrelator

Laser

Sample

θ

i

ωk f

f

Heterodyne Beam

Fig. 16.3. Schematic of a heterodyne scattering experiment.

In the heterodyne experiment, a beam splitter is used to take part of the unscattered beamand route it directly to the PMT where it is mixed with the scattered field. The electric field asso-ciated with this unscattered light is called the local-oscillator field, ELO(t), and of course whatthe PMT sees is ELO(t) + E (1)(q, t) and in fact, the square of this quantity, hence,

I (t)

I0

∝|ELO(t) + E (1)(q, t)|2 = |ELO(t)|2 + ELO(t)E (1)*(q, t) + ELO(t)*E (1)(q, t) + |E (1)(q, t)|2,

(16.40)

Notice that since ELO(t) and E (1)(q, t) both contain a factor of exp(iω i t), this high frequencytime dependence completely cancels out. What is left are the slower phenomena contained inδ ε f ,i(q, t). Of course, if we were to simply signal average the intensity, the second and thirdterms would average out, leaving us with only the intensity of the heterodyning beam and a unin-teresting correction due to the scattering. To avoid this, the intensity is passed to an electronicdevice called an autocorrelator that computes

C(t) ≡ ∫T

0

ds

TI (s + t)I (s) = ⟨I (t)I (0)⟩. (16.41)

When we use Eq. (16.40) in (16.41), 16 terms are obtained. Those linear in E (1)(q, t) average tozero and many of the other ones are independent of the correlation time t, and thus, simplyappear as a base-line in the data. Moreover, |ELO | >> |E (1)|, and the leading order contribution tothe time dependent part of C(t) can be shown to be

C(t)∝2 Re[E*LO(t)ELO(0)⟨E (1)(q, t)E (1)*

(q, 0)⟩]∝2 Re ⟨δ ε f ,i(q, t)δ ε *f ,i(q, 0)⟩;

(16.42)

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which is exactly the same quantity whose Fourier transform was being measured in the opticalfiltering scheme.

16.3.2.2. Homodyne Mixing

In a homodyne detection scattering experiment, the instantaneous intensity from the detec-tor is correlated and

C(τ ) ≡T →∞lim ∫

T

−T

dt

2TI (t + τ )I (t)

is measured. By replacing the time average by an equilibrium ensemble average, and assumingthat the scattering volume probed in the experiment is large enough to contain a large number ofstatistically independent scattering volumes, show that

C(τ ) = ⟨|Esc |2⟩2 + |⟨Esc(0)*Esc(τ )⟩|2,

where Esc is the scattered electric field (at the appropriate scattering wave vector). How doesthis compare with the heterodyne result? The assumption that you are probing a large number ofindependent scattering volumes allows us to use the central limit theorem. This implies that thescattered field, Esc, will have a Gaussian distribution. In particular, if Ai , i = 1, 2, . . . are Gauss-ian distributed (with zero mean for simplicity), note that

⟨A1 A2 A3 A4⟩ = ⟨A1 A2⟩⟨A3 A4⟩ + ⟨A1 A3⟩⟨A2 A4⟩ + ⟨A1 A4⟩⟨A3 A2⟩.

16.4. Appendix: div, grad, and curl

The basic definitions are all obtained from the vector operator

∇ ≡ ex

∂∂x

+ ey

∂∂y

+ ez

∂∂z

, (16.1)

where ei is a unit vector in the "i" direction. Thus,

div f ≡ ∇ ⋅ f =∂ f x

∂x+

∂ f y

∂y+

∂ fz

∂z, (16.2)

grad Φ ≡ ∇Φ = ex

∂Φ∂x

+ ey

∂Φ∂y

+ ez

∂Φ∂z

, (16.3)

and

curl f ≡ ∇ × f =

ex

∂∂x

f x

ey

∂∂y

f y

ez

∂∂z

fz

= ex

∂ fz

∂y−

∂ f y

∂z

− ey

∂ fz

∂x−

∂ f x

∂z

+ ez

∂ f y

∂x−

∂ f x

∂y

. (16.4)

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With these one can prove many useful equalities; e.g.

curl (grad Φ) = 0, (16.5)

div (curl f) = 0, (16.6)

div (fΦ) = Φ(div f) + f ⋅ grad Φ, (16.7)

and

curl (curl f) = grad (div f) − ∇2f. (16.8)

etc.*.

*For a much larger list, see, e.g., I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products

(Academic Press. New York, 1980), Ch. 10.

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17. Problem Sets

Note that the due dates are last year’s. This year’s will be announced in class and on the web

site.

17.1. Problem Set 1

DUE: Friday, January 23, 2015

1. For a gas of hard spheres, the virial expansion is

β P

ρ= 1 +

n=1Σ Bnηn = 1 + 4η + 10η2 + 18. 365η3 + 28. 24η4 + 39. 5η5 + 55. 5η6+. . . ,(16.1.1)

where

η ≡π σ 3 ρ

6

is the packing fraction (σ is the hard sphere diameter). What does η = 1 signify?

In Eq. (1.1), the first 4 terms were evaluated analytically, while the remaining ones involvenumerical integration. Carnahan and Starling noticed that if the coefficients were roundedoff to the nearest integer, they then could be computed from an expression of the form

Bn = a1n2 + a2n + a3, n = 1, 2, . . . (1.2)

What are a1, a2, and a3? How well does the Carnahan and Starling form give the exactBn? Assuming that Eq. (1.2) holds for all n, resum the virial expansion and plot the result-ing equation of state for 0 ≤ η ≤ 1. (It turns out that this agrees remarkably well with com-puter simulation over the entire fluid branch of the equation of state).

2. In extending the ideas of the last problem, Barboy and Gelbart assumed that

π σ 3 β P

6=

∞

n=1Σ B*

nηn =∞

n=1Σ C*

n yn,

where y ≡ η/(1 − η).

Express the C*n in terms of the B*

n. HINT:

(1 + x)−n =∞

j=0Σ

n + j − 1

n − 1

(−x) j .

The series in y is called the y-expansion. By using the exact virial coefficients given inEq. (2.1) above, evaluate C*

1 , C*2 , C*

3 , and C*4 . Compare your result with the Carnahan-

Starling form.

3. Use the general expression we obtained for the pressure of a liquid and the leading-orderform for the pair correlation function, i.e.,

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g(r) ≈ e−β u(r),

to show that the second viral coefficient is the same as that derived in class.

4. Near a critical point, experiment shows that

S(q)∝κ

1 + (qξ )2,

where

κ ≡ −1

V

∂V

∂P

T ,N

→ |T − Tc |−γ ,

is the isothermal compressibility, and where

ξ → |T − Tc |−ν .

The critical exponents γ and ν are universal. What does this imply for the form of g(r)?

5. Prove the convolution theorem for Fourier transforms. If

C(r) ≡ ∫ dr1 f (r − r1)g(r1),

and if the Fourier transforms are defined as

C(k) ≡ ∫ dr eik⋅rC(r), etc. ,

then

C(k) = f (k)g(k).

Note that this has several interesting implications for experiments, most notably in the wayfrequency filters or auto-correlators work.

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Problem Set 2 -129- Chemistry 593

17.2. Problem Set 2

DUE: Friday, February 5, 2010

1. In this problem you will write and run a program for solving the OZ-HNC equation in a onecomponent system. Recall that in the HNC closure,

g(r) = e−β u(r)+h(r)−c(r), (1.1)

or equivalently,

c(r) = e−β u(r)+s(r) − 1 − s(r), (1.2)

where s(r) ≡ h(r) − c(r). Note that while h(r) and c(r) can each be discontinuous at a disconti-nuity in the potential (or in practical terms at a place where the potential changes rapidly), s(r)will be continuous. You can see this by noting the Ornstein-Zernike equation in terms of s(r)becomes

s(r) = ρ ∫ dr1 c(|r − r1|)h(r1).

When the OZ equation is Fourier transformed, and the result is expressed in terms of s(q)it is easy to show that:

s(q) =ρ c2(q)

1 − ρ c(q). (1.3)

For spherically symmetric functions, f (r), show that

f (q) =4πq ∫

∞0

dr r sin(qr) f (r). (1.4)

If we approximate the integral in Eq. (1.4) by a sum, i.e., by using the trapezoid rule, it followsthat

f (k) =4π ∆r2

k∆q

L−1

j=1Σ sin( j k ∆r ∆q) j f ( j), (1.5)

where ∆r and ∆q are the spacings of the grid points in r- and q-space, respectively. The sum inEq. (1.5) can be performed efficiently by using a fast-Fourier-transform (FFT) sin routine (I’veput the source code for one on the course web site). Most sin FFT routines have ∆r∆q = π /L andrequire that L must be a power of 2.

The inverse Fourier transform has a similar form, specifically,

f (r) =4π

(2π )3r ∫∞0

dq q sin(qr) f (q), (1.6)

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or on a grid as

f (k) =4π ∆q2

(2π )3k∆r

L−1

j=1Σ sin( j k ∆r ∆q) j f ( j) (1.7)

Since discontinuities cause problems for discrete Fourier transform methods, we’ll try to avoidthem as much as possible; in particular, if there is a discontinuity in the potential, make sure thatyour grid is chosen so that the discontinuity is at one of the grid points exactly.

We now turn to the numerical solution of the HNC equations. We will solve the equationsiteratively (this is known as the Picard method). Proceed as follows:

a) Choose some initial sold(r) (e.g., zero).

b) Use the closure, Eq. (1.2), to calculate c(r).

c) Use the FFT routine to compute c(k) on a grid.

d) Use the OZ equation to compute a new value of s(k).

e) Again use the FFT to invert the transform and obtain snew(r).

f) Compare snew(r) and sold(r) at each grid point. If

|snew(r) − sold(r)| ≤ AER + RER × |snew(r)|

at each grid point you’re done. Typically AER can be zero, and RER can be 10−4). If not,you must replace sold by snew and go back to step b. If you’re using a programming languagethat supports pointers, e.g., C or C++, they can be used to handle the switch, avoiding thecomputer overhead incurred by copying.

In actual practice, it turns out that this simple iteration doesn’t converge very rapidly, can haveoscillations, and worst of all, be unstable. You can improve the convergence in several ways:

a) Introduce Broyles mixing; specifically, before repeating the iteration, let

snew(r) = α snew(r) + (1 − α )sold(r)

where the mixing parameter, α , is around 0.5.

b) You can slowly change one of the physical parameters, e.g., density or temperature. Forexample, if you want a high density solution, you can first obtain a low density one, and useit as the initial guess for the high density one. You may have to do this in several stages.

Use your program to obtain correlation functions and structure factors for a hard spherefluid, with packing fraction, φ = 0.1, 0.2, 0.3, and 0.4. The packing fraction is defined asφ = π σ 3 ρ /6, where σ is the hard-sphere diameter. Use an FFT grid containing 2048 points, andchoose the upper cutoff in the r integrals to correspond to about 10 σ .

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17.3. Problem Set 3 DUE: Friday, February 12, 2010

In this problem you will write and run a Monte Carlo program for studying the two dimensionalIsing model. In the Ising model, each lattice point has a spin-like variable that can take on values±1. If the interactions are limited to the nearest-neighbor lattice positions, the Hamiltonian forthis problem is written as

H = −kBTJn.nΣ si s j ,

where the sum is restricted to nearest-neighbor pairs of spins and J is a dimensionless couplingconstant. For J > 0 we hav e ferromagnetic interactions, in that the interactions favor parallelspin alignment, while for J <) we have antiferromagnetic interactions. The Ising model is thesimplest model describing an interacting spin system, but can also be used as a crude model ofliquids, polymers, etc.. For example, by interpreting sites with si = 1 as occupied and those withsi = −1 as empty, we hav e a simple lattice model for a liquid.

Use the Metropolis Monte Carlo scheme and write a program to study the average magne-tization, and susceptibility of a two dimensional square lattice,

< S > and χ ≡< S2 > − < S >2 ,

respectively, where

S ≡iΣ si .

(Show that < S2 > − < S >2=< (S− < S >)2 > is proportional to the magnetic susceptibility of aspin system). Assume periodic boundary conditions to handle the edge spins.

Use your program to calculate < S > and χ for J=0, .2, ln(1 + √2) / 2 = 0. 44068679, 1, and 10.Do this for Monte Carlo runs of 106, 108, and 109 spin flips, on a 50x50 lattice. What are theexact results for J = 0?

HINTS:

1) In the standard C, there is a built-in random number generator called rand().

2) Carry out the Monte Carlo moves by randomly selecting a lattice point and testing whetheryou should flip the spin. Note that the energy change for flipping a single spin is

2J(si−1, j + si+1, j + si, j−1 + si, j+1)si, j ,

where si, j is the value of the spin before the flip.

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17.4. Problem Set 4

DUE: Last day of class.

1. In hydrodynamics (or more generally in non-equilibrium thermodynamics), the time depen-dence of conserved quantities obey continuity equations. For example, the mass density, ρ(r, t)obeys a continuity equation,

∂ρ(r, t)

∂t

r

= −∇ ⋅ j(r, t),

where j(r, t) is the mass current. Alternately, we can integrate the continuity equation over anarbitrary volume and use the divergence theorem to show that

∂∂t

V

∫ dr ρ(r, t) = −A

∫ dA n ⋅ j(r, t), (1.1)

where V is an arbitrary volume, A is its surface, and n is an outward normal on A.

Show that the classical Liouville equation for a system having N degrees of freedom canbe written in the form as a probability continuity equation on a 2N dimensional phase space.What is the probability current? Discuss the phase space analog of Eq. (1.1).

2. Show that the classical Liouville operator,

iL ≡iΣ ∂H

∂ pi

∂∂qi

−∂H

∂qi

∂∂ pi

,

is Hermitian; i.e., show that

∫ dX N A*(X N )LB(X N ) = ∫ dX N [LA(X N )]* B(X N ),

where * denotes complex conjugation, and where A and B are assumed to vanish whenever pi orqi → ±∞. Does your answer change if we let dX N → dX N feq(X N ), where feq(X N ) is the equi-librium canonical distribution function?

3. Show that equivalent results are obtained if one averages a mechanical at time t, A((X N (t))),under the initial non-equilibrium distribution function, f (X N , t = 0), or averages a mechanicalquantity at phase point X N under the distribution function at time t; i.e., show that

∫ dX N f (X N , t = 0)A((X N (t))) = ∫ dX N f (X N , t)A(X N ).

Remember that f must obey Liouville’s equation.

4. By using Liouville’s equation, show that the classical entropy,

S ≡ −kB ∫ dX N f (X N , t) ln(( f (X N , t))),

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is a constant of the motion. Prove the same result quantum mechanically. How can this be?

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5. In our discussion of linear response theory, cf. Eq.(1.13), we introduced the response function,

χ B,A(t) ≡i

h−⟨[B(t), A]⟩, (5.1)

where t ≥ 0 and where B(t) and A are Heisenberg Hermitian operators, both assumed to averageto zero,

a) Show that

⟨AB(t)⟩ = ⟨B(t)A⟩* (5.2a)

and hence,

χ B,A(t) = −2

h−Im(⟨B(t)A⟩), (5.2b)

where "Im" and "*" denote the imaginary part and complex conjugate, respectively.

b) Show that

ε →0+lim

εε 2 + x2

= π δ (x),

where δ (x) is the Dirac δ -function and x is real

c) By using the energy representation, show that

χ B,A(t) =βQ

d

dt n,mΣ eiω n,m t Bn,m A*

n,m

e−β En − e−β Em

β h− ω n,m

, (5.3a)

where n, m denote energy eigenstates and ω n,m ≡ (En − Em)/h−. Next, carry out thehalf-sided Fourier transform and show that

χ B,A(ω ) =βQ n,m

Σ Bn,m A*n,m


β h− ω n,m

ω n,m

ω − ω n,m + iε(ω − ω n,m)2 + ε 2

, (5.3b)

where we have let ω → ω − iε , ε > 0, to guarantee that the Fourier transform inte-gration converges.

d) By using the result in part b), take the ε → 0 + limit and show that

χ B,A(ω ) = iω β π

1 − eβ h−ω

β h− ω n,m

Σ e−β En

QBn,m A*

n,mδ (ω − ω n,m)

+βQ n,m

Σ Bn,m A*n,m


β h− ω n,m

ω n,m

ω − ω n,m

. (5.4)

e) Assume that B = A and show that Eq. (5..4) becomes

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χ A,A(ω ) ≡ χ ′A,A(ω ) + i χ ′′

A,A(ω ),

where the real and imaginary parts of χ A,A(ω ) are

χ ′A,A(ω ) =

βQ n,m

Σ |An,m |2


β h− ω n,m

ω 2n,m

ω 2 − ω 2n,m

(5.5a)

and

χ ′′A,A(ω ) = β π ω

1 − eβ h−ω

β h− ω n,m

Σ e−β En

Q|An,m |2δ (ω − ω n,m), (5.5b)

respectively. Show that ω χ ′′A,A(ω ) ≤ 0.

f) Consider the regular time correlation function, C A,A(t) ≡ ⟨A(t)A⟩, where t can bepositive or neg ative. Show that

C A,A(ω ) = 2πn,mΣ e−β En

Q|An,m |2δ (ω − ω n,m) (5.6)

and hence,

χ ′′A,A(ω ) =

1 − eβ h−ω

2h−C A,A(ω ). (5.7)

Thus there is a simple relation and the correlation function, although note that theyare not the same!

g) Discuss these results in the context of the fluctuation dissipation theorem.

6. An inner product, (A, B), (a generalization of the usual dot product) must satisfy three proper-ties:

i) (A, B) = (B, A)*, where * denotes complex conjugation.

ii) For any scalars β and γ , (A, β B + γ C) = β (A, B) + γ (A, C)

iii) (A, A) > 0 unless A = 0.

a) Show that an equilibrium average can be used to define an inner product.

b) Use your result to prove that for any set of complex numbers and times, ci and ti , respec-tively, any time correlation function must satisfy

i, jΣ c*

i c j < A*(ti)A(t j) > > 0.

c) Finally, by appropriately choosing the ci and ti , show that the time Fourier transform of atime auto-correlation function must be real and positive. This is known as Bochner’s theo-rem. How does the result generalize for multivariate correlations (i.e., matrices of timecorrelation functions)?

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7. For a classical one component system interacting via pairwise additive potentials, show thatthe Fourier transform of the energy density, Ek(t) obeys the following continuity equation:

dEk(t)

dt= ik ⋅ JE (k, t),

where the energy current (flux) is given by

JE (k, t) ≡j

Σ eik⋅r j(t)

p2j(t)

2m j

+1

2 j′≠ jΣ u j, j′((r j, j′(t)))

↔1 +

1

2 j′≠ jΣ r j, j′(t)F j, j′(t)

1 − e−ik⋅r j, j′(t)

ik ⋅ r j, j′(t)

⋅p j

m j

.

Note that this form of the energy current is also obtained for mixtures. Discuss the physical sig-nificance of the different terms. Also note the similarity of the terms that explicitly depend onthe forces and those found in the stress tensor, i.e., in

↔τ (k, t) =j

Σ eik⋅r j(t)

p jp j

m j

+1

2 j′≠ jΣ r j, j′(t)F j, j′(t)

1 − e−ik⋅r j, j′(t)

ik ⋅ r j, j′(t)

.

What happens if we replace the individual particle momenta by the average momenta? How canyou interpret this result?

8. A broken symmetry system exhibits long ranged static correlations. For example, in a ferro-magnet below the Curie temperature

< Mk ⋅ M−k >≈R

k2as k → 0,

where Mk is the Fourier transform of the magnetization density and R is a constant. What doesthis imply for the real space decay of the magnetization-magnetization correlation function? Themagnetization is not conserved mechanically, nonetheless, magnetization fluctuations exhibitslow diffusive time dependence. Use linear response theory to derive the form of the averageequation of motion for the magnetization, and give a Green-Kubo form for the diffusion constant.

9. In a homodyne detection scattering experiment, the instantaneous intensity from the detectoris correlated and

C(τ ) ≡T →∞lim ∫

T

−T

dt

2TI (t + τ )I (t)

is measured. By replacing the time average by an equilibrium ensemble average, and assumingthat the scattering volume probed in the experiment is large enough to contain a large number ofstatistically independent scattering volumes, show that

C(τ ) =< |Esc |2 >2 +| < Esc(0)*Esc(τ ) > |2,

where Esc is the scattered electric field (at the appropriate scattering wave vector). How doesthis compare with the heterodyne result? The assumption that you are probing a large number ofindependent scattering volumes allows us to use the central limit theorem. This implies that the

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scattered field, Esc, will have a Gaussian distribution. In particular, if Ai , i = 1, 2, . . . are Gauss-ian distributed (with zero mean for simplicity), note that

< A1 A2 A3 A4 >=< A1 A2 >< A3 A4 > + < A1 A3 >< A2 A4 > + < A1 A4 >< A3 A2 > .

david ronis mcgill universityronispc.chem.mcgill.ca/ronis/chem593/course_pac.pdf · statistical...

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