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.^[1][2] 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

• ${\displaystyle q_{i,i-1}=i\,\lambda }$ for i = 1, 2, ..., N
• ${\displaystyle q_{i,i+1}=(N-i\,)\lambda }$ for i = 0, 1, ..., N – 1

and equilibrium distribution ${\displaystyle \pi _{i}=2^{-N}{\tbinom {N}{i}}}$.

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

${\displaystyle H(t)=-\sum _{i}P(X(t)=i)\log \left({\frac {P(X(t)=i)}{\pi _{i}}}\right),}$

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