Major Concepts Lecture #11 Rigoberto Hernandez. TST & Transport 1

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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 1

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 Nobel Prize in Chemistry But note that Callen & Welton [PRB 83, (1951)] proved the FDT for microscopic disturbances TST & Transport 2

3 Onsager s Regression Hypothesis Spontaneous fluctuations: Relaxation of a disturbance: correlation function Onsager s hypothesis: TST & Transport 3

4 Onsager s Regression Hypothesis Examples: Velocity autocorrelation function: Relaxation in chemical kinetics: C(t)/C(0) TST & Transport K.M. Solntsev, D. Huppert, N. Agmon, J. Phys. Chem. A 105(2001)5868 4

5 Onsager s Regression Hypothesis Limiting behavior of the correlation function: Note: TST & Transport 5

6 Fluctuation Dissipation Theorem Equilibrium average value of a variable A: Given a small (microscopic) disturbance: such that calculate initial value TST & Transport 7

7 Fluctuation Dissipation Theorem Average value of a dynamical variable A(t): But TST & Transport 8

8 Fluctuation Dissipation Theorem Average value of A(t): because TST & Transport 9

9 Result: Fluctuation Dissipation Theorem If ΔH = fa(0) then ΔA (t) = βfc(t) Onsager s regression hypothesis TST & Transport 10

10 Fluctuation Dissipation Theorem Given a small (microscopic) disturbance: This is equivalent to the Onsager s Regression Hypothesis when the latter is applied to small perturbations. TST & Transport 11

11 Chemical Kinetics Simple Kinetics Phenomenology Master Equation Detailed Balance 1 E.g.: apparent rate for isomerization : τ rxn = kab + k BA Microscopic Rate Formula Relaxation time Plateau time TST & Transport 13

12 Rates The rate is: 1 E.g., in the apparent rate for isomerization : τ rxn = k AB + k BA k(0) is the transition state theory rate After an initial relaxation, k(t) plateaus (Chandler): the plateau or saddle time: t s k(t s ) 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

Transition State Theory Objective: Calculate reaction rates Obtain insight on reaction mechanism Eyring, Wigner,

No dynamical recrossings of TST Keck,Marcus,Miller,Truhlar, Others.

.. PODS 2-Dimensional non-recrossing DS Full-Dimensional Non-Recrossing Surfaces Miller, Hernandez developed good

13 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 (Marcus: Science 256 (1992) 1523) TST & Transport 15

14 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

15 Fluxes, Affinities & Transport Coefficients, II Suppose a Solution: With conserved quantities, U, and N s solutes Entropy Production, S(U,N s ) Recognize the affinities γ as the S-conjugate variables: $ γ E = S ' & ) % E ( N s $ γ Ns = S ' & ) % N s ( E = 1 T = µ S T Out of equilibrium local fields, ρ S (r,t) and u(r,t) 2 nd Law of thermodynamics implies that differences in affinities drives fluxes: j E j N ( r,t) = L EE γ E ( r,t) + L EN γ N ( r,t) ( r,t) = L NE γ E ( r,t) + L NN γ N ( r,t) TST & Transport 18

16 Fluxes, Affinities & Transport Coefficients, I The Transport Equation in this Linear Response Regime for solutes are: j E j N ( r,t) = L EE γ E ( r,t) + L EN γ N ( r,t) ( r,t) = L NE γ E ( r,t) + L NN γ N ( r,t) Limits: Constant N Fourier s law for heat conduction j E = λ T with thermal conductivity : λ = L EE T 2 µ = k B T ln ρ Constant T & nearly dilute Fick s Law: j N = D ρ with diffusion constant : D = L NN k B ρ In general, Temperature and Particle gradients can drive each other! The coefficients L ij are the Onsager Coefficients The Onsager Reciprocity Relations simply say that L is diagonal, i.e., that L ij = L ji for all I and j. Diagonal terms capture the usual spread or diffusion of the corresponding property directly TST & Transport 19

17 The Diffusion Equation, I Mass transport equation: j N = L NN γ N Mass conservation equation (aka Equation of Continuity) without sources or sinks: The general Diffusion Equation: The usual Diffusion Equation: at low solute concentration, γ N = k B ln ρ$ assuming Fick s Law, D = L k & % NN B ρ '& ρ r,t ρ t = j N ( ) t ρ( r,t) t = L NN γ N ( r,t) = D 2 ρ( r,t) TST & Transport 20

18 The Diffusion Equation, II In the Diffusion Equation: at low solute concentration, γ N = k B ln ρ$ assuming Fick s Law, D = L & NNk B % ρ '& we observed that D is proportional to L NN 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, D = λk B T ρ( r,t) t = D 2 ρ( r,t) TST & Transport 21

19 The Diffusion Equation, III The Diffusion Equation: ρ( r,t) t In Fourier space w.r.t. wave vector k: ( ) ρ k,t Which can be solved for a given BC, e.g., ρ k,0 : ( ) = N S And then inverse Fourier transformed: ρ( r,t) = N S ( 4πDt) 3 / 2 exp r r 0 ' 2Dt % & ( ) 2 t = D 2 ρ( r,t) ( ) = Dk 2 ρ k,t ρ( k,t) = N S exp( k 2 Dt) ( ) * 1 dr r r 3 0 ( ) 2 ρ( r,t) = 2Dt TST & Transport 22

Major Concepts Langevin Equation

Major Concepts Langevin Equation Model for a tagged subsystem in a solvent Harmonic bath with temperature, T Friction & Correlated forces (FDR) Markovian/Ohmic vs. Memory Chemical Kinetics Master equation