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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)
8843 Lecture #11 —Rigoberto Hernandez
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
8843 Lecture #11 —Rigoberto Hernandez
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:
8843 Lecture #11 —Rigoberto Hernandez
TST & Transport
5
Onsager’s Regression Hypothesis • Limiting behavior of the correlation function:
Note:
8843 Lecture #11 —Rigoberto Hernandez
TST & Transport
8843 Lecture #11 —Rigoberto Hernandez
TST & Transport 7
• Equilibrium average value of a variable A:
• Given a small (microscopic) disturbance:
such that calculate initial value
Fluctuation Dissipation Theorem
8843 Lecture #11 —Rigoberto Hernandez
TST & Transport 8
• Average value of a dynamical variable A(t):
• But
Fluctuation Dissipation Theorem
8843 Lecture #11 —Rigoberto Hernandez
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• Average value of A(t):
Fluctuation Dissipation Theorem
because
8843 Lecture #11 —Rigoberto Hernandez
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• Result:
• If then
– Onsager’s regression hypothesis
Fluctuation Dissipation Theorem
€
ΔA (t) = βfC(t)
€
ΔH = − fA(0)
8843 Lecture #11 —Rigoberto Hernandez
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.
Fluctuation Dissipation Theorem
8843 Lecture #11 —Rigoberto Hernandez
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
8843 Lecture #11 —Rigoberto Hernandez
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
8843 Lecture #11 —Rigoberto Hernandez
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…
8843 Lecture #11 —Rigoberto Hernandez
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:
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€
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
8843 Lecture #11 —Rigoberto Hernandez
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€
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:
8843 Lecture #11 —Rigoberto Hernandez
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€
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,
8843 Lecture #11 —Rigoberto Hernandez
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:
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€
∂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