Ehrenfest model

The Ehrenfest model (or dog-flea model) 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

and equilibrium distribution π i = 2 − N ( N i ) .

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

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