Ehrenfest model

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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.


  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 [1] 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.