Nonlinear Ehrenfest's Urn Model

The Ehrenfest's urn model (sometimes also called Ehrenfest's flea
model) has played an important role in
clarifying the foundations of statistical mechanics, providing
an interpretation of irreversibility in a statistical
manner. The model is defined by $N$ balls distributed in two urns
(or boxes) 1 and 2, such that at each discrete instant of time $s$,
a ball is chosen at random and moved from the box in which it
is found to the other box. At the beginning of the 20th century,
such a simple model was useful in explaining the heat exchange between two bodies
at unequal temperatures, where the temperatures are mimicked by
the number of balls in each box, and the heat exchange becomes a random
process. In the present work the Ehrenfest's
urn model is modified by introducing nonlinear
terms in the associated transition probabilities. It is shown that
these modifications lead, in the continuous limit, to a
Fokker-Planck equation characterized by two competing
diffusion terms, namely, the usual linear one, as well as a
nonlinear diffusion term, typical of anomalous diffusion.
By considering a generalized H-theorem,
the associated entropy is calculated, resulting in
a sum of Boltzmann-Gibbs and Tsallis entropic forms.
It is shown that the stationary state of the associated
Fokker-Planck equation satisfies precisely the same equation
obtained by extremization of the entropy.
Moreover, the effects of the nonlinear contributions on the
entropy production phenomenon are also analyzed.