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3.1: Modeling the Approach to Equilibrium

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Equilibrium

A thermodynamic system typically consists of an enormously large number of constituent particles, a typical ‘large number’ being Avogadro’s number, NA=6.02×1023. Nevertheless, in equilibrium, such a system is characterized by a relatively small number of thermodynamic state variables. Thus, while a complete description of a (classical) system would require us to account for O(1023) evolving degrees of freedom, with respect to the physical quantities in which we are interested, the details of the initial conditions are effectively forgotten over some microscopic time scale τ, called the collision time, and over some microscopic distance scale, , called the mean free path1. The equilibrium state is time-independent.

The Master Equation

Relaxation to equilibrium is often modeled with something called the master equation. Let Pi(t) be the probability that the system is in a quantum or classical state i at time t. Then write

dPidt=j(WijPjWjiPi) .

Here, Wij is the rate at which j makes a transition to i. Note that we can write this equation as

dPidt=jΓijPj ,

where

Γij={Wijif ijkWkjif i=j ,

where the prime on the sum indicates that k=j is to be excluded. The constraints on the Wij are that Wij0 for all i,j, and we may take Wii0 (no sum on i). Fermi’s Golden Rule of quantum mechanics says that

Wij=2π|i|ˆV|j|2ρ(Ej) ,

where ˆH0|i=Ei|i, ˆV is an additional potential which leads to transitions, and ρ(Ei) is the density of final states at energy Ei. The fact that Wij0 means that if each Pi(t=0)0, then Pi(t)0 for all t0. To see this, suppose that at some time t>0 one of the probabilities Pi is crossing zero and about to become negative. But then Equation ??? says that ˙Pi(t)=jWijPj(t)0. So Pi(t) can never become negative.

Equilibrium distribution and detailed balance

If the transition rates Wij are themselves time-independent, then we may formally write

Pi(t)=(eΓt)ijPj(0) .

Here we have used the Einstein ‘summation convention’ in which repeated indices are summed over (in this case, the j index). Note that

iΓij=0 ,

which says that the total probability iPi is conserved:

ddtiPi=i,jΓijPj=j(PjiΓij)=0 .

We conclude that ϕ=(1,1,,1) is a left eigenvector of Γ with eigenvalue λ=0. The corresponding right eigenvector, which we write as Peqi, satisfies ΓijPeqj=0, and is a stationary ( time independent) solution to the master equation. Generally, there is only one right/left eigenvector pair corresponding to λ=0, in which case any initial probability distribution Pi(0) converges to Peqi as t, as shown in Appendix I (§7).

In equilibrium, the net rate of transitions into a state |i is equal to the rate of transitions out of |i. If, for each state |j the transition rate from |i to |j is equal to the transition rate from |j to |i, we say that the rates satisfy the condition of detailed balance. In other words,

WijPeqj=WjiPeqi.

Assuming Wij0 and Peqj0, we can divide to obtain

WjiWij=PeqjPeqi .

Note that detailed balance is a stronger condition than that required for a stationary solution to the master equation.

If Γ=Γt is symmetric, then the right eigenvectors and left eigenvectors are transposes of each other, hence Peq=1/N, where N is the dimension of Γ. The system then satisfies the conditions of detailed balance. See Appendix II (§8) for an example of this formalism applied to a model of radioactive decay.

Boltzmann’s H-theorem

Suppose for the moment that Γ is a symmetric matrix, Γij=Γji. Then construct the function

H(t)=iPi(t)lnPi(t) .

Then

dHdt=idPidt(1+lnPi)=idPidtlnPi=i,jΓijPjlnPi=i,jΓijPj(lnPjlnPi) ,

where we have used iΓij=0. Now switch ij in the above sum and add the terms to get

dHdt=12i,jΓij(PiPj)(lnPilnPj) .

Note that the i=j term does not contribute to the sum. For ij we have Γij=Wij0, and using the result

(xy)(lnxlny)0 ,

we conclude

dHdt0 .

In equilibrium, Peqi is a constant, independent of i. We write

Peqi=1Ω,Ω=i1H=lnΩ .

If ΓijΓji, we can still prove a version of the H-theorem. Define a new symmetric matrix

¯WijWijPeqj=WjiPeqi=¯Wji ,

and the generalized H-function,

H(t)iPi(t)ln(Pi(t)Peqi) .

Then

dHdt=12i,j¯Wij(PiPeqiPjPeqj)[ln(PiPeqi)ln(PjPeqj)]0 .


This page titled 3.1: Modeling the Approach to Equilibrium is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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