Ehrenfest model

The Enrenfest model of diffusion was proposed by Paul Ehrenfest to explain the second law of thermodynamics. The model considers N particles in two containers. Particles independently change container at a rate λ. If X(t) is defined to be the number of particles in one container at time t, then it is a birth-death process with transition rates

* $q\_\{i\; i-1\}\; =\; i\; lambda$ for i=1,2,...,N
* $q\_\{i\; i+1\}\; =\; (N-i)\; lambda$ for i=0,1,...,N–1

and equilibrium distribution $pi\_i\; =\; 2^\{-N\}\; binom\; Ni$.

It can be shown that if the initial system state is not equilibrium, then the entropy given by

:$H(t)\; =\; -sum\_\{i\}\; P(X(t)=i)\; log\; left(\; frac\{P(X(t)=i)\}\{pi\_i\}\; ight)$

is monotonically increasing. This is a consequence of the convergence to the equilibrium distribution.

References

* F.P. Kelly Reversibility and Stochastic Networks (Wiley, Chichester, 1979) ISBN 0471276014 [http://www.statslab.cam.ac.uk/~frank/BOOKS/kelly_book.html] pp.17-20
* "Ehrenfest model of diffusion." Encyclopædia Britannica (2008)