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\title{General Mathematics and Computational Science I}
\author{Exercise 21}
\date{December 1, 2005}
\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 \textbf{Extra credit.} Write, if you can, a computer program to
simulate the experiment.

\item How would you characterize the behavior of $D(t)$ for short and
for long times?  How can you relate this behavior to the Loschmidt and
the Zermelo paradox?  (A qualitative answer is sufficient.)

\item What could be the entropy for this system with its associated
irreversible time evolution?  

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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