Mathematical economics

The Ehrenfest model (or dog-flea model) 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 ${\pi }_i=2^{-N}(\genfrac{}{}{0ex}{}{N}{i})$.

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.

References

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• 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.