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 $\pi _{i}=2^{-N}{\tbinom {N}{i}}$.

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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• 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.