Quantum Statistical Mechanical Derivation of Generalized HydrodynamicEquationsBaldwin Robertson Citation: Journal of Mathematical Physics 11, 2482 (1970); doi: 10.1063/1.1665414 View online: http://dx.doi.org/10.1063/1.1665414 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/11/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantum statistical derivation of the hydrodynamics of a plasma model J. Math. Phys. 26, 2324 (1985); 10.1063/1.526816 Some General Inequalities in Quantum Statistical Mechanics J. Math. Phys. 12, 1123 (1971); 10.1063/1.1665707 Derivation of the Generalized Boltzmann Equation in Quantum Statistical Mechanics J. Math. Phys. 7, 1039 (1966); 10.1063/1.1704996 Statistical Mechanical Derivation of Onsager's Equation for Dielectric Polarization J. Chem. Phys. 22, 1806 (1954); 10.1063/1.1739924 The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics J. Chem. Phys. 18, 817 (1950); 10.1063/1.1747782

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JOURNAL OF MATHEMATICAL PHYSICS VOLUME II, NUMBER 8 AUGUST 1970

Quantum Statistical Mechanical Derivation of Generalized Hydrodynamic Equations

BALDWIN ROBERTSON National Bureau of Standards, Washington, D.C. 20234

(Received 29 December 1969)

Differential conservation equations are derived for the masso, momentum-, and energy-density operators for a I-component simple fluid of Bose or Fermi particles with arbitrary pairwise interactions. These equations are used in a statistical mechanical derivation of exact equations of motion for the expectations of these operators. The equations of motion are coupled to equations relating these expecta­tions to the local temperature, chemical potential, and fluid velocity. The coupled equations are closed in the sense that the expectations and their thermodynamic conjugates listed above are the only un­knowns, although some of the dependence in the eq~tions on the conjugates is expressed only implicitly. The equations of motion are memory-retaining nonlocal generalizations of the classical hydrodynamic equations and apply to a normal fluid arbitrarily far from equilibrium. The formalism is not carried as far as has the corresponding classical formalism because the local equilibrium expectation of the momentum density here does not equal the fluid velocity times the expectation of the mass density as is true in classi­cal statistical mechanics.

1. INTRODUCTION the Wigner density to transform Irving and Kirk-In this paper generalized hydrodynamic equations wood's7 classical derivation into a quantum derivation.

valid for fluids far from equilibrium are derived from FrohlichB used an expansion about total equilibrium, quantum statistical mechanics. The equations are but did not give expressions for the coefficients in this closed since the currents are expressed as functionals expansion. Without attempting to derive hydro­ofthe local thermodynamic conjugate variables, which dynamic equations, Kadanoff and Martin9 used them themselves are functionals of the mass, momentum, to determine the space and time dependence of corre­and energy densities. The system considered is a 1- lation functions. Hohenberg and MartiniO discussed component simple fluid of Bose or Fermi particles superfluids by making assumptions in order to close with arbitrary pairwise interactions. This system is their hydrodynamic equations. None of these authors often taken as a model for liquid 4He or 3He. However, used statistical mechanics to derive closed equations except for a brief comment on superfluids, this paper valid for fluids far from equilibrium. is concerned only with normal fluids and hence not Morill derived hydrodynamic equations by ex-with 4He below 2.17 K. panding the statistical density operator about local

Although some of the material in this paper has equilibrium. He obtained expressions for the trans­appeared previously in a general formalism,I.2 a self- port coefficients as time integrals of correlation func­contained derivation applicable to quantum fluids is tions,I2-I4 but only after removing the thermodynamic presented here for greater clarity. The generalized forces from the integrals. IS This can be done only when hydrodynamic equations to be derived are identical there is a wide separation in time scales. His formalism in form to the corresponding equations derived from is closely related to an approximation of the one to be classical statistical mechanics. The latter equations presented. may be transformed into those outlined without The remainder of this paper in outline is as follows. derivation by Richardson3 and derived in detail by The Hamiltonian and the operators corresponding to Piccirelli4 for a fluid of interacting classical particles. the observed densities are defined in Sec. 2. Then However, the present formalism is not carried as far exact operator equations, in the form of the classical as Piccirelli's because the local equilib:·ium expecta- hydrodynamic equations, are written down in Sec. 2, tion of the momentum density j<; not as simple in with their derivation given in Appendix A. Local quantum theory as in Classical theory. As far as the thermodynamic variables such as temperature and present paper goes, it gives an efficient, exact, quan- chemical potential are defined in Sec. 3 using the local tum statistical derivation of closed equations, pre- equilibrium statistical density operator. Then exact viously widely believed to be impossible to derive closed generalized hydrodynamic equations, which are exactly. the expectations of the above-mentioned operator

There are not many papers devoted to the derivation equations, are written down in Sec. 3, with their of hydrodynamic equations from quantum statistical derivation given in Appendix B. Further steps, omitted mechanics. Born and Green5 used a density-matrix in this paper, are discussed in Sec. 4, along with a hierarchy to obtain equations similar in form to brief comment on extending the formalism to include hydrodynamic equations. Irving and ZwanzigG used superfluids.

2482

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QUANTUM STATISTICAL DERIVATION OF HYDRODYNAMIC EQUATIONS 2483

2. OPERATOR EQUATIONS

The Hamiltonian of the 1-component simple fluid to be considered here is

Je == f d3rtp+(r) ( - :: V2 + VI(r)

+ t f d3r' tp+(r') V2(r - r')tp(r'») tp(r), (2.1)

where tp+ and tp are Bose or Fermi particle creation and annihilation operators. Each particle has the external potential energy VI , and each pair of particles has the interaction potential energy V2 , which may be any given functions.

The operators usually assumed to describe this fluid are the mass-, momentum-, and energy-density operators

Pm(r) == tp+(r)mtp(r),

p,,(r) == tp+(r)ptp(r), and

Peer) == tp+(rMtp(r),

respectively. Here

p == (1i/2i)(V - V),

(2.2)

(2.3)

(2.4)

(2.5)

where the arrows over the gradient operators indicate in which direction they operate: to the right on tp(r) or to the left on tp+(r). Also

It == p2 + VI(r) + tfd3r'tp+(r')V2(r - r')tp(r'), (2.6) 2m

where the p2 again operates to the right and left. This definition of Peer) with the operator p2 is used because it is Hermitian, because its integral over all space equals the Hamiltonian (2.1), and because it simplifies some equations later on.

In classical hydrodynamics, the time derivatives of the mass, momentum, and energy densities equal the divergences of the corresponding currents plus an external force term in the momentum equation. These are called differential conservation laws. Now the commutator ofiJe/1i with an operator corresponds to a time derivative. This commutator with the operators (2.2)-(2.4) can be expressed as divergences because total mass, momentum, and energy are conserved (except for the effect of the external force).

These commutators are calculated in Appendix A, and the resulting operator equations are

(i/Ii)[Je, Pm(r)] = -V· pir), (2.7)

(i/Ii)[Je, pir)] = Pm(r)FI(r)/m - A· jp(r), (2.8) and

(i/Ii)[Je, p.(r)] = -V· j.(r), (2.9)

where the momentum current is given by

jir) == tp+(r)(pp/m )tp(r) + t f dSr' f d3r" f" dr'"

x r5(r - r"')F2(r' - r")tp+(r")tp+(r')tp(r')tp(r"),

(2.10) and the energy current is given by

j.(r) == tp+(r)l[p/m, It]+tp(r)

+ .!.. fd3r'fd3r"!r,, dr'" oCr - r",)F lr' - r") 4m r'

• tp+(r")tp(r')(p' + p")tp(r')tp(r"). (2.11) Here,

(2.12)

is the external force on a particle, and

F2(r' - r") == -VV2(r' - r") (2.13)

is the force on a particle at r' exerted by another particle at r".

The line integral in Eqs. (2.10) and (2.11) may be taken along any curve between the points r' and r", except as restricted by the following. The expectation of the currents must always be zero for all r for which the expectation of the mass density Pm(r) is zero. Con­sistent with this restriction, one way of removing the above ambiguityI6 is to use the shortest curve between r' and r" that remains inside the region where the expectation of Pm(r) is nonzero. For most r' and r", the curve is.a straight line, and the currents (2.10) and (2.11) are then analogous to Richardson's3.17 for classical statistical mechanics.

Infinite-series expressions for the currents jp and j. have been given by Irving and Kirkwood7 for classical statistical mechanics, and by Grossmann18 for quan­tum statistical mechanics. Higher-order terms in these expressions can be dropped only if the interparticle force F2 is short range. Richardson's currents, as well as the currents (2.10) and (2.11), both agree with these series expansions. Kugler's currents,19 however, do not, agree with the series expansions even in the short­range limit, and their expectations are not zero where the expectation of Pm(r) is zero.

3. CLOSED EQUATIONS

The expectations of Eqs. (2.7)-(2.9) are differential conservation laws similar in form to the equations of classical hydrodynamics. However, the expectations of the currents jp and j. on the right are as yet un­knowns, and there are more unknowns than equations. In order to obtain the hydrodynamic equations, it is necessary to express these currents in terms of the temperature and other thermodynamic variables,

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2484 BALDWIN ROBERTSON

which themselves must be expressed in terms of the expectations of the mass-, momentum-, and energy­density operators (2.2)-(2.4). Then the equations will be closed.

The expectations of the mass-, momentum-, and energy-density operators are given by

(Pm(r»t == Tr [Pm(r)p(t)], (3.1)

(pp(r»t == Tr [pp(r)p(t)], (3.2) and

(Pe(r»t == Tr [Pe(r)p(t)], (3.3)

where p(t) is the statistical density operator satisfying the Liouville equation. These expectations are the un­knowns whose values are to be found by deriving the closed equations of motion they satisfy and then inte­grating these equations of motion.

In order to derive these equations, we introduce new variables (J(r, t), fl(r, t), and vCr, t), which are defined to be functionals of the expectations (3.1)-(3.3) as follows. Introduce the local equilibrium statistical density operator

aCt) == _1_ exp [-jd3r{J(r, t) Z(t)

X (Pe(r) - fl(: t) Pm(r) - v(r, t) • pp(r») 1 (3.4)

where for normalization

Z(t) == Tr {exp [ - j d3r{J(r, t)

X (p.(r) - fl(~ t) Pm(r) - vCr, t) • pir») ]}.

(3.5)

The (J(r, t), fl(r, t), and v(r, t) are to be chosen to satisfy

Tr [Pm(r)a(t)] = (Pm(r»t, (3.6)

Tr [pir)a(t)] = (pp(r»t, (3.7) and

Tr [p.(r)a(t)] = (pir»t, (3.8)

where the functions on the right are already defined in Eqs. (3.1)-(3.3). Equations (3.4)-(3.8) are not to be considered here as expressions for (Pm), (pp), and (Pe), but are coupled nonlinear integral equations for {J, fl, and v as unknowns. The functional dependence of {J, fl, and v on (Pm), (pp ), and (Pe) just defined is a time­independent one; {J, fl, and v depend upon t only because they depend upon (Pm), (pp), and (Pe), which themselves depend upon t. The multipliers {Jfl, {Jv, and {J are called the thermodynamic conjugates of the

expectations (Pm), (pp), and (Pe)' Here (J(r, t) is the local temperature, fl(r, t) + tmv(r, t)2 is the local chemical potential, and vCr, t) is the local fluid velocity.2o.21

Equations (3.4)-(3.9) are just definitions and are to be used even for large deviations from equilibrium, as in a shock wave, for example. No assumption is made here that the fluid remains in any sense close to equilibrium. The statistical density operator p(t) is as­sumed to equal the local equilibrium statistical density operator aCt) only at the initial time t = O. This initial condition is reasonable for a fluid initially constrained away from equilibrium. 21 The fluid would be initially in equilibrium only if (J and fl were con­stants and v were zero.

All of the traces to be calculated in the following will involve a(t) rather than pet). So, for convenience, let angular brackets have the definition

(A)t == Tr [Aa(t)], (3.9)

where A may be any quantum-mechanical operator. Because of the definition of (J, ft, and v, this definition is consistent with Eqs. (3.6)-(3.8). Of course, the local equilibrium statistical density operator aCt) does not satisfy the Liouville equation. As a result, the expecta­tion of an arbitrary operator A is not given by Eq. (3.9), but must be calculated as a trace of A times pet), which does satisfy the Liouville equation.

The expectations of Eqs. (2.7)-(2.9), including the currents jp and je, are calculated exactly in Appendix B, and the resulting closed equations of motion are

O(Pmo~»t = -V. (Pr(r»t> (3.10)

o(pp(r»t = F1(r)(Pm(r»t _ V . (Mr»t ot m

and

+ V • I'dt' J d3r'[ KIIlr, t, r', t') :V'{J(r', t')v(r', t')

- Kpe(r, t, r', t'). V'{J(r', t')], (3.11)

ot

+ V .fdt'f d3r'[ Kelr, t, r', t'):V'{J(r', t')v(r', t')

- K .. (r, t, r', t') • V' (J(r', t')], (3.12)

where the kernels are correlation functions given by

Kpp(r, t, r', t') == (jp(r)T(t, t')(1 - P)jp(r'»t' , (3.13)

Kpe(r, t, r', t') == (jp(r)T(t, t')(1 - P)j.(r'»t" (3.14)

Kep(r, t, r', t') == (i.(r)T(t, t')(1 - P)jlr'»t" (3.15)

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QUANTUM STATISTICAL DERIVATION OF HYDRODYNAMIC EQUATIONS 2485

and

Ke.(r, t, r', tf) == (je(r)T(t, t')(l - P)je(r'»t'. (3.16)

Here, the operator P is related to a projection operator with respect to the weight (J.22 It automatically subtracts out an average of the current j to its right, so that the correlation functions can approach zero for large t - t'. The operator T gives the unitary23 transforma­tion on the operator to its right, advancing it in time, but with the hydrodynamic motion projected out at each instant by the operator P. This dependence of Ton P, given by Eq. (B5), is essential if the corre­lation functions are to become zero and remain zero after a time that is short compared with hydro­dynamic times.24 More generally, the correlation functions may decay in about the same time as the expectations (3.1 )-(3.3), and P is still essential.

Equations (3.10)-(3.12) are memory-retaining non­local generalizations of the classical hydrodynamic equations. In a near-equilibrium approximation, the correlation functions (3.13)-(3.15) satisfy reciprocity relations. 25 However, no approximations are made in the present paper. The first terms on the right of Eqs. (3.10)-(3.12) are called reversible terms since they do not directly change the entropy.26 The time integral terms are called irreversible terms since they do change the entropy. The reversible terms describe only the macroscopic flow, while the irreversible terms describe the microscopic dissipative effects due to the inter­action V2 •

Equations (3.4)-(3.16) are the desired closed equa­tions of motion for (Pm(r»t, (p/r»t, (Pe(r»t, ~(r, t), p.(r, t), and vCr, t). These are the only unknowns in the equations. However, the dependence of the kernels on ~, p.; and v is stated only implicitly in the definition of P and T given in Appendix B. A method for extracting this dependence approximately is discussed in Sec. 4.

4. DISCUSSION

Equations (3.10)-(3.l2) are memory-retaining non­local generalizations of the classical hydrodynamic equations. The advantages of this formalism are: (1) The equations of motion are closed since the corre­lation functions (3.13)-(3.16) are functionals of the local temperature, local chemical potential, and local fluid velocity, which themselves are given in Eqs. (3.4)-(3.8) as functionals of the mass, momentum, and energy densities. (2) All traces are to be calculated using the local equilibrium statistical density operator a(t) as in Eq. (3.9). This is physically desirable and is easier than calculating traces using pet), which, as a solution to the Liouville equation, is linear in the initial condition and thus contains the initial condition

explicitly. In the present formalism, the initial con­dition is contained only implicitly through the initial values of the macroscopic unknowns. (3) The equa­tions of motion are exact and apply to systems arbitrarily far from equilibrium. They apply even when the classical hydrodynamic equations do not, e.g., near a critical point where a memory-retaining nonlocal theory is necessary. (4) The equations reduce to the classical hydrodynamic equations in the appropriate limits, and the resulting expressions for the transport coefficients are independent of the order of taking these limits, as is discussed below.

Equations (3.4)-(3.16) are identical in form to generalized hydrodynamic equations derived from classical statistical mechanics, except that expressions in them involve traces over operators rather than inte­grals over functions of phase. However, as will be discussed, the dependence of these expressions on the local velocity vCr, t) is more complicated in quantum theory than it is in classical theory. Hence, approxi­mations appear to be necessary here, although they are not necessary at this step in the classical derivation.

In the classical derivation,4 the reversible currents (pp), (jp), and (je) become (Pm>v, (Pm>vv + (jp)+, and (Pe)v + (jp)+ • v, respectively. Here the plus indicates that, in the local equilibrium density (J, the v term is dropped and p. is replaced by p. + tmv2• These results can be proved4 by translating one of the momentum integrals by mv. When this is done on the irreversible terms, the thermodynamic forces V' ~v and V' ~ become V'v and V'~. Thus the equations take a form closely paralleling the classical hydrodynamic equations.

In the quantum derivation, the reversible currents have additional terms,27 which are much too compli­cated to write here, but which vanish in the limit of smaI11i2~ or in the limit of small V x v. Also, except in one of these limits, the vV ~ term cannot be removed from the thermodynamic force V ~v. Thus, it appears to be necessary, at this point, to take the approxima­tion in which terms containing 1i2~ and V x v are dropped.

In the short-memory local limit, the correlation functions become sharp like t5 functions, so that the thermodynamic forces V'v and V' f3 can be removed from the integrals, which then become transport coeffi­cients. 28 These correlation functions must still con­tain the operator P both explicitly and through the dependence given by Eq. (B5), so that they will be­come zero and remain zero after a time that is short compared with hydrodynamic times. 24 This operator P does not appear in the Green-Kubo12- 14 expressions for transport coefficients. As a result, in their expres­sions the integral over an infinite volume must be

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2486 BALDWIN ROBERTSON

performed before the integral over the infinite-time interval in order to obtain a nonzero result for the transport coefficients.29 However, because in the present formalism the operator P projects out the hy­drodynamic motion at each instant in the time development of the correlation functions, the order of integration here is not important, and the space inte­gral here need not be infinite.

The final step in deriving the equations of motion is to obtain explicit expressions for the kernels in terms of quantities that can be calculated. For linear deviations from equilibrium, this can be done with a continued fraction expansion.30 A simple method of truncating this kind of expansion has been used to predict a nuclear magnetic resonance line shape in excellent agreement with experiments.31 The application of this technique to fluids is limited at the present by the difficulty of calculating the moments, which involve traces over <1.

For superfluids, additional variables and additional equations of motion coupled to Eqs. (3.10)-(3.12) appear to be necessary. A derivation of these exact closed equations has not yet been accomplished except for zero interaction V2 • If V2 is zero, the extra vari­ables are the expectations of ljJ+(r) and ljJ(r), and Eqs. (A2) and (A3) are used along with Eqs. (2.7)-(2.9). Terms linear in ljJ+(r) and ljJ(r) must be added onto the exponent in Eqs. (3.4) and (3.5), where the multipliers in these terms are determined by Eqs. (3.6)-(3.8) plus two new equations involving the expectations of ljJ+(r) and ljJ(r). The result is that extra terms are added to Eqs. (3.10)-(3.12) coupling these equations to two new equations of motion that are the expectations of Eqs. (A2) and (A3). By use of a canonical transforma­tion whose exponent is linear in ljJ+(r) and ljJ(r), the particle-nonconserving terms added to the exponent in Eqs. (3.4) and (3.5) can be transformed away and the reversible terms in Eqs. (3.10)-(3.12) become the usual sum of super and normal terms appearing in the phenomenological theory.32 This is a beautifully simple derivation of 2-fluid generalized hydrodynamic equa­tions for the noninteracting Bose fluid. However, the irreversible terms here are not zero and cannot be approximated in a short-memory local limit. If Va is not zero, the problem with the irreversible terms is removed, but expressions containing Va become un­pleasant when the above canonical transformation is performed. At the present it is not known how to correct this.

ACKNOWLEDGMENT The author is grateful to Dr. R. A. Piccirelli for

many helpful conversations in which both the general nature of this paper and many details were discussed.

APPENDIX A

In this appendix, Eqs. (2.7)-(2.11) are derived using

[ljJ(r), ljJ(r')]:;:: = 0, [ljJ(r), ljJ+(r')]:;:: = d(r - r'),

[ljJ+(r), ljJ+(r')]:;:: = 0, (A1)

where the minus sign denotes commutator for bosons and the plus sign denotes anticommutator for fer­mlOns.

For either bosons or fermions, Eqs. (2.1) and (AI) give

( /i2

[.le, ljJ(r)] = - - - V2 + V1(r) 2m

+ J dar' ljJ+(r') V2(r - r')VJ{r'») ljJ(r) (A2)

+ f d3r' ljJ+(r') V2(r - r')ljJ(r') ), (A3)

where the upper (lower) equation is most easily derived by expanding the commutator on the left into two terms and using Eqs. (AI) on the second (first) term only. No integration by parts is necessary.

Equations (A2) and (A3) along with the identities

[.le, ljJ+AljJ] = [.le, ljJ+]AljJ + ljJ+[.le, A]ljJ

+ ljJ+A[.le, ljJ], (A4)

V2 - V2 = - (\7 + V) . C - V), (AS) and

[p, A] = -ili(VA)

give

[.le, ljJ+(r)AljJ(r)]

= iliV· {ljJ+(r)t[p/m, Al+ljJ(r)}

+ ljJ+(r)(.le + p2 + Vl(r) 2m

(A6)

+ J d3r'ljJ+(r')V2(r - r')VJ{r'), A )ljJ(r), (A7)

where A may be any linear operator. Notice that the V in [p, A]+ does not operate on A.

Equation (A7) is used as follows. Let A = im/Ii to get Eq. (2.7). Let A = ip/Ii and use Eq. (A6) to get

(i/ Ii)[.le, PII(r)]

= Pm(r)F1(r)/m - V • [ljJ+(r)(pp/m)VJ{r)]

-f d3r'F2(r' - r)tp+(r)tp+(r')VJ{r')VJ{r), (A8)

where FI(r) and F2(r) are defined by Eqs. (2.12) and (2.13). Finally, let A = M/Ii and use Eqs. (2.5) and

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QUANTUM STATISTICAL DERIVATION OF HYDRODYNAMIC EQUATIONS 2487

(A6) to get

(i/Ii)[Je, p.(r)] = -V· {fjJ+(rH[p/m, .4]+fjJ(r)}

- 1- fd3r'F2(r' - r) 2m

• fjJ+(r)fjJ+(r')(p + p')fjJ(r')fjJ(r),

(A9)

where the p' term comes from the [Je, .4] term in Eq. (A7).

The last terms in Eqs. (AS) and (A9) are of the form

where

f d3r'!(r, r'), (AlO)

fer', r) = -fer, r'). (All)

As a result, the integral of these terms over all r is zero, and the terms can be written as divergences

f d3r'f(r, r')

= V -J d3rJ d3r"f"dr"'tb(r - r''')f(r', r"). (AI2)

This is easily proved by letting the yo operate on the b to become - VIII, thus permitting the rill line integral to be performed.

When Eq. (AI2) is applied to Eqs. (AS) and (A9) , Eqs. (2.8) and (2.9) result, where the currents are defined by Eqs. (2.10) and (2.11).

APPENDIX B

In this appendix, Eqs. (3.10)-(3.16) are derived from the Liouville equation. Although this derivation is a special case of a previously published general formal­ism,1.2 it is presented here in a self-contained form for convenience in the present application.

The Liouville equation is

a;~t) = -iLp(t),

where the Liouville operator is defined by

LA == [Je, A)/Ii

(B1)

(B2)

for any quantum-mechanical operator A. Here Je is ~he Hamiltonian (2.1). The initial condition for pet) IS assumed to be

p(O) = 0'(0). (B3)

where A may be any quantum-mechanical operator. Also, it is convenient to introduce another operator T(t, t') defined by

aT(t t') 0;' = T(t, n[1 - P(t')]iL (BS)

with the initial condition

T(t, t) = l. (B6)

The operators pet) and T(t, t'), like L, operate to their right on quantum-mechanical operators. Both P and Tare functionals of fl, fl, and v.

Equations (BS), (Bl), and (B4), and the chain rule for calculating the total derivative of O'(t') give

o{T(t, t')[p(t') - O'(t')]}/ot'

= - T(t, t')[1 - P(t')]iLO'(t').

Integrate this over t' from 0 to t and use Eqs. (B6) and (B3) to get

pet) = O'(t) - fdt'T(t, t')[l - P(t')]iLO'(t'). (B7)

This expresses the nonequilibrium statistical density operator satisfying Eqs. (Bl)-(B3) as a functional of fl, fl, and v.

A different expression for the nonequilibrium statistical density operator pet) as a function of O'(t) has be~n giv~n by Zubarev.33 His expression, although exact, IS not In a form that is convenient for obtaining exact equations of motion formally similar to hydro­dynamic equations. His resulting approximate ex­pressions for the currents do not involve the operator pet) and, hence, do not have the advantages listed following Eq. (3.16) above.

The application of Eq. (B7) is aided by the identity

LA = J: dxA"'(L log A)A1-"', (B8)

which may be proved for' an arbitrary quantum­mechanical A by integrating dA"JeAl-"/dx over x from o to l. Equations (B8), (3.4)"and (2.7)-(2.9) give

iLO' = f d3rt{ V· j. - flV ~pp + V· ~Fl - VV:jpJO',

(B9)

In order to express the expectations of the operators where the bar over an operator is defined by

jp and j. in terms of just the desired unknowns, it is (1 convenient to introduce an operator pet) defined by A == Jo O'(t)'" AO'(t)-'" dx - (A)t (B10)

(B4) for any quantum-mechanical operator A. The last term in Eq. (B1O) drops out of Eq. (B9) because

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2488 BALDWIN ROBERTSON

Tr (aL log a) IS zero for any statistical density oper­ator a and any Hamiltonian Je in Eq. (B2).

When Eq. (B9) is used in Eq. (B7), the terms in Pm and Pp drop out. This is proved as follows. The well-known34 rule for differentiating an operator gives

and

__ ba-,,(~t)_ = b(J(r, t)p(r, t)

Pm(r)a(t)

m

baCt) - t .£(J( ) ( ) = pp(r)a( ), u r, t v r, t

(Bll)

(B12)

where the bar is defined in Eq. (BIO) and a is given in Eq. (3.4). Equations (B4) , (BIt), and (BI2) and the chain rule for calculating a total derivative give

P(t)Pm(r)a(t) = Pm(r)a(t) (B13) and

P(t)pir)a(t) = pp(r)a(t). (B14)

Because of Eqs. (B13) and (BI4), the Pm and pp terms drop out of [1 - P(t)]iLa(t), where iLa is given by Eq. (B9).

When this result is used along with Eq. (B7) to calculate the expectations of jp and je in Eqs. (2.7)­(2.9) and when Eqs. (3.1)-(3.3) and (BI) are used on the other terms, Eqs. (3.10)-(3.16) follow after an integration by parts in the space integral on the right. In this integration the surface term vanishes because the current operators, by definition, vanish outside the volume containing the fluid.

1 B. Robertson, Phys. Rev. 144, 151 (1966). 2 B. Robertson, Phys. Rev. 160, 175 (1967); 166,206 (1968), Er-

ratum. 8 J. M. Richardson, J. Math. Anal. Appl. 1, 12 (1960). • R. A. Piccirelli, Phys. Rev. 175, 77 (1968). 5 M. Born and H. S. Green, Proc. Roy. Soc. (London) A191, 168

(1947). 8 J. H. Irving and R. W. Zwanzig, J. Chern. Phys. 19, 1173 (1951). 7 J. H. Irving and J. G. Kirkwood, J. Chern. Phys.18, 817 (1950). 8 H. Frohlich, Physica 37, 215 (1967). 9 L. P. Kadanoff and P. C. Martin, Ann. Phys. (N.Y.) 24, 419

(1963). . 10 P. C. Hohenberg and P. C. Martin, Ann. Phys. (N.Y.) 34, 291

(1965). 11 H. Mori, J. Phys. Soc. Japan 11,1029 (1956); Phys. Rev. 112,

1829 (1958); 115, 298 (1959). 12 M. S. Green, J. Chern. Phys. 20,1281 (1952); 22,398 (1954). 13 R. Kubo, J. Phys. Soc. Japan 12, 570 (1957); R. Kubo, M.

Yokota, and S. Nak3jima, J. Phys. Soc. Japan 12, 1203 (1957).

14 R. W. Zwanzig, Ann. Rev. Phys. Chern. 16, 67 (1965). 15 G. V. Chester, Rept. Progr. Phys. 26, 452 (l963). 18 A similar ambiguity results from the possibility of adding to the

currents the curl of an arbitrary operator that vanishes wherever the expectation of the mass density vanishes. These ambiguities suggest that only the divergences of the currents (rather than the currents themselves) should be used. The currents themselves are used here in order to retain the formal similarity with classical hydrodynamics. Even so, the currents always have V· operating on them in the formalism to follow.

17 This formal similarity is most apparent with Eqs. (28) and (29) of Ref. 4.

18 S. Grossmann, Z. Physik 191,103 (1966). The results of interest above are given in Eqs. (24) and (35), where S is defined in Eq. (4a). Most of this paper concerns systems with nonlocal interactions, for which the continuity equation (2.7) above is not valid. No examples of such a system are given. Also, Grossmann states that any density can be made to satisfy a differential conservation law (with the source term zero), provided only that a suitable current is used. He states that this is always possible even if the total volume integral of the density is not conserved. However, for densities for which the source term is not naturally zero, this requires a current whose con­tribution to a surface integral exactly equals the volume integral of the source. Such a current is not physical. For example, the density pp in Eq. (2.8) above has the source F1Pm/m, which is not a natural divergence. The author thanks Dr. Grossmann for bringing this paper to his attention.

19 A. Kugler, Z. Physik 198, 236 (1967). 20 See the discussion following Eq. (17) of Ref. 1. 21 See the references given in the left column of p. 176 of Ref. 2. 22 Appendix B of Ref. 2. 23 Equation (B4) of Ref. 1. 24 The author thanks Dr. R. A. Piccirelli and Dr. W. C. Mitchell

for pointing out this important advantage of the present formalism. 25 Appendix C of Ref. 1. 28 Footnote 18 of Ref. 2. .7 The explicit form of these terms is, of course, a consequence of

the Definitions (2.3) and (2.4) assumed for the form of the momen­tum- and energy-density operators. It is not known whether a dif­ferent definition would make the additional terms vanish. However, with the present definitions, it is just the additional term in the reversible current (p,) that gives a nontrivial result in the quantum theory of the equilibrium diamagnetism of a charged fluid, where v is replaced by the vector potential A and !A2 is subtracted from {-t.

28 This is discussed in detail on p. 181 of Ref. 2. Notice that the correlation functions are not completely determined because any solenoidal current operator that vanishes outside the volume con­taining the fluid can be added to ip or i. without violating the defini­tion of these currents. The equations of motion (3.11) and (3.12) do not have this ambiguity because of the V operators, which can be made to operate on ip and i. after an integration by parts. In order to eliminate the ambiguity from the transport coefficient [e.g., Eq. (30) of Ref. 2], the r dependence must be averaged over the volume containing the fluid as in Eqs. (38) on p. 405 of Ref. 12. Then, only the volume integrals of the currents appear, and these depend only on the divergences of the currents and on the currents being zero outside, as can be seen by integrating rV • j by parts.

29 See, e.g., R. W. Zwanzig, J. Chern. Phys. 40, 2527 (1964). 30 H. Mori, Pro gr. Theoret. Phys. (Kyoto) 34, 399 (1965). 31 B. Robertson, Bull. Am. Phys. Soc. 12, 1141 (1967). 32 I. M. Khalatnikov, Introduction to the Theory of SuperJluidity

(Benjamin, New York, 1965). 33 D. N. Zubarev, Dokl. Akad. Nauk SSSR 164, 537 (1965)

[Sov. Phys. DokI. 10, 850 (1966)]. See also T. N. Khazanovich, Mol. Phys. 17,281 (1969), Eqs. (3.1)-(3.3).

84 See, e.g., Appendix A of Ref. 1.

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