Q. 5(Ehrenfest model).The eminent physicist Paul Ehrenfest (1880-1933) proposed the following simple model for the exchange of gas molecules between two containers that are connected by a narrow opening. There areNgas molecules in total across the two containers. The number of molecules in the first container at timenis denotedXn. The basic Ehrenfest model works as follows: in every time step, one of theNmolecules is selected randomly (independently from the past, and with equal probability for every molecule) and moved to the opposite container than it is currently in.

a. Show that{Xn}is a Markov chain, and find its transition probabilities. b.{Xn}is not ergodic. Why not?

To make the Ehrenfest model ergodic, we consider the following modification of the dynamics. As before, in every time step, one of theNmolecules is selected randomly. We now put this molecule in one of the two containers at random, independently of the molecule that was chosen; each container is chosen with equal probability. Thus there is a nonzero probability that the ball that was selected will not be moved. From now on, we will assume that{Xn}is defined according to this alternative mechanism.

c. Find the transition probabilities of the Markov chain{Xn}.

1. {Xn}is ergodic. Why?
2. What is the limiting probability asn of findingjmolecules in the first con- tainer? [Hint: in view of the way in which the model is defined, it is natural to guess that after a long time, this model behaves as though we had putNgas molecules independently in one of the two containers with equal probabilities.]