major concepts - school of chemistry and biochemistryrh143/courses/wordup/13a/lec/11.pdf · major...

8843 Lecture #11 —Rigoberto Hernandez

TST & Transport 1

Major Concepts •  Onsager’s Regression Hypothesis

–  Relaxation of a perturbation –  Regression of fluctuations

•  Fluctuation-Dissipation Theorem –  Proof of FDT –  & relation to Onsager’s Regression Hypothesis –  Response Functions

•  Kinetics & TST •  Phenomenology & Transport

–  C.f. BH Sections 9.1 & 9.2 –  Entropy Production, Affinities & Onsager Reciprocity

Relations –  The Diffusion Equation (driven by density fluctuations) –  Cahn-Hillard Equation (density and energy fluctuations)


TST & Transport 2

Onsager’s Regression Hypothesis •  Concepts:

–  An equilibrium system has fluctuations –  An equilibrium system which is instantaneously in an

fluctuation looks like a non-equilibrated system that must relax to equilibrium

•  Onsager: –  “The relaxation of macroscopic non-equilibrium

disturbances is governed by the same laws as the regression of spontaneous microscopic fluctuations in an equilibrium system.”

–  1968 Nobel Prize in Chemistry –  But note that Callen & Welton [PRB 83, 34-40 (1951)]

proved the FDT for microscopic disturbances


TST & Transport 3

Onsager’s Regression Hypothesis •  Spontaneous fluctuations:

– correlation function

•  Relaxation of a disturbance:

Onsager’s hypothesis:

4

Onsager’s Regression Hypothesis •  Examples:

C(t)

/C(0

)

Velocity autocorrelation function:

K.M. Solntsev, D. Huppert, N. Agmon, J. Phys. Chem. A 105(2001)5868

Relaxation in chemical kinetics:


TST & Transport

5

Onsager’s Regression Hypothesis •  Limiting behavior of the correlation function:

Note:


TST & Transport


TST & Transport 7

•  Equilibrium average value of a variable A:

•  Given a small (microscopic) disturbance:

such that calculate initial value

Fluctuation Dissipation Theorem


TST & Transport 8

•  Average value of a dynamical variable A(t):

•  But



TST & Transport 9

•  Average value of A(t):


because


TST & Transport 10

•  Result:

•  If then

– Onsager’s regression hypothesis


€

ΔA (t) = βfC(t)

€

ΔH = − fA(0)


TST & Transport 11

•  Given a small (microscopic) disturbance:

•  This is equivalent to the Onsager’s Regression Hypothesis when the latter is applied to small perturbations.



TST & Transport 13

Chemical Kinetics

•  Simple Kinetics—Phenomenology – Master Equation – Detailed Balance – 

•  Microscopic Rate Formula – Relaxation time – Plateau time €

E.g. : apparent rate for isomerization : τ rxn−1

= kAB + kBA

Rates

•  The rate is:

–  k(0) is the transition state theory rate

–  After an initial relaxation, k(t) plateaus (Chandler): •  the plateau or saddle time: ts •  k(ts) is the rate (and it satisfies the TST Variational Principle)

–  After a further relaxation, k(t) relaxes to 0

•  Other rate formulas: –  Miller’s flux-flux correlation function –  Langer’s Im F


TST & Transport 14

€

E.g., in the apparent rate for isomerization : τ rxn

−1 = kAB + kBA

(Marcus: Science 256 (1992) 1523)

Transition State Theory • Objective:

•  Calculate reaction rates •  Obtain insight on reaction mechanism

• Eyring, Wigner, Others.. 1.  Existence of Born-Oppenheimer V(x) 2.  Classical nuclear motions 3.  No dynamical recrossings of TST

• Keck,Marcus,Miller,Truhlar, Others... •  Extend to phase space •  Variational Transition State Theory •  Formal reaction rate formulas

• Pechukas, Pollak... •  PODS—2-Dimensional non-recrossing DS

•  Full-Dimensional Non-Recrossing Surfaces •  Miller, Hernandez developed good action-angle variables at the

TS using CVPT/Lie PT to construct semiclassical rates •  Jaffé, Uzer, Wiggins, Berry, Others... extended to NHIM’s, etc


TST & Transport 15

Fluxes, Affinities & Transport Coefficients, I

•  (Barat & Hansen, Section 9.1) •  Local Thermal Equilibrium (LTE)

– Allows for separation between mesoscopic subsystems in LTE and nonequilibrium macroscopic variables

– Defines, e.g., ρ(r,t) and T(r,t) •  We now aim to construct (Non-Eq)

phenomenological evolution equations based on LTE at the mesoscale…


TST & Transport 17

Fluxes, Affinities & Transport Coefficients, II

•  Suppose a Solution: – With conserved quantities, U, and Ns solutes – Entropy Production, S(U,Ns) – Recognize the affinities ! as the S-conjugate

variables:

– Out of equilibrium local fields, ρS(r,t) and u(r,t) –  2nd Law of thermodynamics implies that

differences in affinities drives fluxes:


TST & Transport 18

€

γ E =∂S∂E$

% &

'

( ) Ns

= 1T

γ Ns=

∂S∂Ns

$

% &

'

( ) E

= −µST

€

jE r, t( ) = LEE∇γE r, t( ) + LEN∇γN r, t( )jN r, t( ) = LNE∇γE r, t( ) + LNN∇γN r, t( )

Fluxes, Affinities & Transport Coefficients, I •  The Transport Equation in this “Linear Response Regime” for solutes are:

•  Limits: –  Constant N… Fourier’s law for heat conduction

–  Constant T & nearly dilute… … Fick’s Law:

–  In general, Temperature and Particle gradients can drive each other!

•  The coefficients Lij are the Onsager Coefficients –  The Onsager Reciprocity Relations simply say that L is diagonal, i.e., that Lij = Lji for all I

and j. –  Diagonal terms capture the usual spread or diffusion of the corresponding property directly


TST & Transport 19

€

jE = −λ∇T with thermal conductivity : λ = LEET 2

€

jN = −D∇ρ with diffusion constant : D = LNNkB

ρ

€

µ = kBT lnρ

€

jE r, t( ) = LEE∇γE r, t( ) + LEN∇γN r, t( )jN r, t( ) = LNE∇γE r, t( ) + LNN∇γN r, t( )

The Diffusion Equation, I •  Mass transport equation: •  Mass conservation equation (aka

Equation of Continuity) without sources or sinks:

•  The general Diffusion Equation:

•  The usual Diffusion Equation:


TST & Transport 20

€

jN = LNN∇γN

€

∂ρ∂t

= −∇jN

€

∂ρ r, t( )∂t

= −∇LNN∇γN r, t( )

€

at low solute concentration, γN = kB lnρ

assuming Fick’s Law, D = LNNkB

ρ

$ % &

' & ⇒

∂ρ r, t( )∂t

= −D∇2ρ r, t( )

The Diffusion Equation, II •  In the Diffusion Equation:

we observed that D is proportional to LNN … •  Is this an accident?

– No, it is an example of the Fluctuation-Dissipation Theorem we already discussed

– That is, it arises from the fact that the mobility λ in response to a drift current is related to the Diffusion constant through the Einstein relation,


TST & Transport 21

€

at low solute concentration, γN = kB lnρ

assuming Fick’s Law, D = LNNkB

ρ

$ % &

' & ⇒

∂ρ r, t( )∂t

= −D∇2ρ r, t( )

€

D = λkBT

The Diffusion Equation, III •  The Diffusion Equation:

•  In Fourier space w.r.t. wave vector k:

•  Which can be solved for a given BC, e.g., :

•  And then inverse Fourier transformed:


TST & Transport 22

€

∂ρ k, t( )∂t

= −Dk2ρ k, t( )

€

∂ρ r, t( )∂t

= −D∇2ρ r, t( )

€

ρ k, t( ) = NS exp −k2Dt( )

€

ρ r, t( ) = NS 4πDt( )−3 / 2exp −

r−r0( )2

2Dt

%

& '

(

) *

€

⇒ 13 dr r − r0( )2∫ ρ r, t( ) = 2Dt

€

ρ k,0( ) = NS

major concepts - school of chemistry and biochemistryrh143/courses/wordup/13a/lec/11.pdf · major...

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