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{\bf Question}

The following is a simple model of the exchange the heat of or gas
molecules between two isolated bodies known as the Ehrenfest
model.

Suppose there are two boxes, labelled 1 and 2, and $d$ balls
labelled $1, 2,\ldots , d.$  Initially some of these balls are in
the box 1 and the remainder are in box 2.  An integer is selected
at random from $1, 2,\ldots ,d,$ and the ball labelled by that
integer is removed from its box and placed in the opposite box.
This procedure is repeated indefinitely with the selections being
independent from trail to trail.  Let $X_n$ denote the number of
balls in box 1 after the $n$-th trial.  Check that $\{X_n\}$ is a
Markov chain and find the 1-step transition probabilities.



\vspace{.25in}

{\bf Answer}

Ehrenfest Diffusion Model

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\begin{picture}(8,5)

\put(0,2){\makebox(0,0)[l]{Box 2}}

\put(0,3){\makebox(0,0)[l]{Box 1}}

\put(1,1.75){\framebox(1,0.5)[c]{$d - j$}}

\put(1,2.75){\framebox(1,0.5)[c]{$j$}}

\put(2,3){\line(2,1){2}}

\put(2,2){\line(2,-1){2}}

\put(4.2,0.7){\makebox(0,0)[l]{2}}

\put(4.2,1.3){\makebox(0,0)[l]{1}}

\put(4.5,0.4){\framebox(1.5,0.5)[c]{$d - j-1$}}

\put(4.5,1.1){\framebox(1,0.5)[c]{$j + 1$}}


\put(4.2,3.7){\makebox(0,0)[l]{2}}

\put(4.2,4.3){\makebox(0,0)[l]{1}}

\put(4.5,3.4){\framebox(1.5,0.5)[c]{$d - j+1$}}

\put(4.5,4.1){\framebox(1,0.5)[c]{$j - 1$}}

\put(2.5,3.7){\makebox(0,0)[l]{$\frac{j}{d}$}}

\put(2.5,1.2){\makebox(0,0)[l]{$\frac{d - j}{d}$}}

\put(7,4.5){\makebox(0,0)[l]{where $\frac{j}{d}$ is the prob.
that}}

\put(7,4){\makebox(0,0)[l]{the ball is taken from box 1}}

\put(7,1.5){\makebox(0,0)[l]{where $\frac{d - j}{d}$ is the prob.
that}}

\put(7,1){\makebox(0,0)[l]{the ball is taken from box 2}}

\end{picture}

Let $X_n =$ number of balls in box 1 after the $n$-th trial.

This depends only on how many balls are in the box before the
trial, and so is a Markov chain.

$P(X_{n+1} = k | X_n = j) = \left\{ \begin{array}{cc} \frac{j}{d}
& {\rm if\ } k = j-1 \\ \frac{d - j}{d} & {\rm if\ \ } k = j + 1
\\ 0 & {\rm otherwise} \end{array} \right.$



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