major concepts - georgia institute of technologyrh143/courses/wordup/14c/lec/18.pdf · major...

6481 Lecture #18 —Rigoberto Hernandez

TCFs, Onsager & FDT 1

Major Concepts •  Nonequilibrium Dynamics

•  Time Correlation Functions –  Diffusion Equation (macro & micro interpretations)

•  Onsager Regression Hypothesis –  Relaxation of a perturbation –  Regression of fluctuations

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

Nonequilibrium Dynamics •  Far-from-equilibrium, systems are different!

–  Doesn’t the solvent average it all out??? •  Cf. Zwanzig’s Topics:

–  Brownian Motion & Langevin Equations –  Fokker-Planck Equations –  Master Equations –  Reaction Rates & Kinetics –  Classical vs. Quantum Dynamics –  Linear Response Theory

•  Use thermodynamic quantities to predict Non-Eq

–  Nonlinearity



See, e.g., R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, 2001)

Time Correlation Functions • 

•  The time correlation function is:

•  The diffusion equation:

•  The diffusion constant is:



Diffusion Equation •  Macroscopic Interpretation

•  Microscopic Interpretation

•  The diffusion constant (a macroscopic quantity) can be connected to the decay in the velocity auto correlation function (a microscopic quantity(:



€

ddt

x 2t

= 2D

€

ddt

x 2(t) = 2 du0

t

∫ v(t)v(0)

Time Dependent Correlation Functions

•  Provide a quantitative description of the dynamics in liquids.

•  Power spectrum is what is measured by many spectroscopic techniques.

•  Linear transport coefficients of hydrodynamics are related to time integrals of time dependent correlation functions.

Why are time dependent correlations functions important?


6 TCFs, Onsager & FDT

Time Dependent Correlation Functions •  Time-correlation function of a dynamical variables A and B is

given by

•  We can also exclude the average values of the dynamical variables and define the correlation function as:

–  Written in this way, as

•  The Fourier transform of the time correlation function is the power spectrum

•  And the Laplace transform is defined as:

•  Autocorrelation functions are real even functions of t and ω 6481 Lecture #18 —Rigoberto Hernandez

7 TCFs, Onsager & FDT



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

•  Limiting behavior of the correlation function:

•  Relaxation of a disturbance:



Onsager’s Regression Hypothesis •  Spontaneous fluctuations:

– correlation function

Onsager’s hypothesis:

10

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:


TCFs, Onsager & FDT



•  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



•  Equilibrium average value of a variable A:


such that calculate initial value




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

•  But




•  Average value of A(t):


because



•  Result:

•  If then

– Onsager’s regression hypothesis


€

ΔA (t) = βfC(t)

€

ΔH = − fA(0)




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


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