Ehrenfest model
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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
- for i = 1, 2, ..., N
- for i = 0, 1, ..., N – 1
and equilibrium distribution .
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.