Ehrenfest model

The Ehrenfest model (or dog-flea model^[1]) of diffusion was proposed by Tatiana and 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) = i 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 $\pi_i = 2^{-N} \tbinom Ni$.

Mark Kac proved in 1947 that if the initial system state is not equilibrium, then the entropy, given by

$H(t) = -\sum_{i} P(X(t)=i) \log \left( \frac{P(X(t)=i)}{\pi_i}\right) ,$

is monotonically increasing (H-theorem). This is a consequence of the convergence to the equilibrium distribution.

References

1. Nauenberg, M. (2004). "The evolution of radiation toward thermal equilibrium: A soluble model that illustrates the foundations of statistical mechanics". American Journal of Physics 72 (3): 313–311. doi:10.1119/1.1632488.

• F.P. Kelly Reversibility and Stochastic Networks (Wiley, Chichester, 1979) ISBN 0-471-27601-4 pp. 17–20
• "Ehrenfest model of diffusion." Encyclopædia Britannica (2008)
• Paul und Tatjana Ehrenfest. Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem. Physikalishce Zeitschrift, vol. 8 (1907), pp. 311-314.