Ehrenfest model

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The Ehrenfest model (or dog–flea model) of diffusion was proposed by Tatiana and Paul Ehrenfest in 1907 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 λ {\displaystyle q_{i,i-1}=i\,\lambda } for i = 1, 2, ..., N q i , i + 1 = ( N − i ) λ {\displaystyle q_{i,i+1}=(N-i\,)\lambda } for i = 0, 1, ..., N – 1 and equilibrium distribution π i = 2 − N ( N i ) {\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 H ( t ) = − ∑ i P ( X ( t ) = i ) log ⁡ ( P ( X ( t ) = i ) π i ) , {\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.