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 rateλ. 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

.

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.

Interpretation of results

. Using Stirling's approximation years. This supposes that while theoretically sure, recurrence to the initial highly disproportionate state is unlikely to be observed.

Bibliography

• Paul and Tatjana Ehrenfest: Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem. Physikalische Zeitschrift, vol. 8 (1907), pp.311–314.

• F.P. Kelly: The Ehrenfest model, in Reversibility and Stochastic Networks. Wiley, Chichester, 1979. pp.17–20.

• David O. Siegmund: Ehrenfest model of diffusion (mathematics). Encyclopædia Britannica.

See also

• Kac ring

• Ornstein–Uhlenbeck process

References