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\title{General Mathematics and Computational Science II}
\author{Exercise 5}
\date{February 15, 2007}
\maketitle

\noindent \emph{Ehrenfest's urn model} (paraphrases from Dorfman's
book).  Consider two urns, I and II, and a bag.  $N$ balls, labeled
$1, \dots, N$, are distributed between the urns.  The bag contains $N$
pieces of paper that carry numbers $1,\dots,N$.  At each time step,
somebody draws a number out of the bag at random, moves the
corresponding ball from its current to the other urn, and then puts
the piece of paper back into the bag.  This is repeated a large number
of times.  Consider the difference
\[
  D(t) = \lvert N_{\text{I}} (t) - N_{\text{II}}(t) \rvert
\]
between the number of balls in urns I and II as a function of time.

\begin{enumerate}

\item In which sense is the behavior of this system time-reversible?
Does it have recurrence?

\item Define an entropy function for this system.

\item Find an equation for $\langle D(t) \rangle$.  What happens for
large $t$? 

(Notice that the chance that a ball will move from the fuller urn to
the emptier urn is always greater than the chance that the opposite
will happen, therefore the system has the tendency to approach an
equilibrium where the balls are distributed equally between the urns.
Fluctuations will also occur, even very large ones.)

\end{enumerate}
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