Ehrenfest model

The Ehrenfest model ( also known as Ehrenfest chain) is a stochastic model that describes the exchange of material between two containers separated by a membrane. The model was first proposed by the Austrian physicist Paul Ehrenfest (1880-1933) and is one of many contributions of physics to develop the mathematical theory of stochastic processes.

The model

In various substances has been observed that the distribution of the substance in such an experiment, in the course of time while strives towards a state of equilibrium, but nevertheless remains exposed after achieving the same always uncontrollable seemingly random fluctuations.

This circumstance was trying to explain the following model:

At the beginning are in two containers together a finite number of particles; about the individual molecules of the substance, of which reside initially in the left and right analog in the container. In each time step, exactly one of these particles is uniformly distributed selected, which changes the container, so that in each step rise and fall exactly by one or.

Mathematically, it is in this random process is a Markov chain with state space and a transition matrix, given by

Mathematical properties

• The above defined Ehrenfest chain has a unique stationary distribution: If the number of particles in the left (or right) container binomial distribution with parameters, that is, so has the same distribution.

• The convergence of the chain against this distribution, however, is not given, since the chain is periodic (notice that in mind that always alternates between even and odd numbers and thus every other time is equal to zero ). This can be circumvented by moving on to the aperiodic version of the chain and the transition matrix for a fixed parameter replaced by the matrix (this is the unit matrix). Interpretation: with probability if the number of particles in the containers unchanged with probability it changes according to the method described above. Wherein the chain is aperiodic and converges against the stationary distribution, which does not change by this modification.