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\title{General Mathematics and ACM II}
\author{Exercise 10}
\date{March 18, 2011}
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\def\d{\mathrm{d}}

\begin{enumerate}
\item Newton's second law of mechanics for a particle of mass $m$
situated at position $x(t)$ moving with velocity $v(t)$ and subject to
a force $F(x(t))$ can be written
\begin{gather*}
  \frac{\d x}{\d t} = v \,, \\
  m \, \frac{\d v}{\d t} = F(x(t)) \,.
\end{gather*}

Use the chain rule of calculus to show that the particle satisfies the
same equation with $t$ replaced by the reversed time $r=-t$ and $v$
replaced by $-v$.

\item Show that in a time-discrete, reversible, system with a finite
number of states any orbit must return to its initial state after a
finite number of steps.

\item In the Kac ring model, $N$ sites are placed around a circle.
The sites are populated with $B$ black balls and $W=N-B$ white balls
at random.  Moreover, $n$ markers are placed on the edges between the
sites at random; the number of black balls just before a marked edge
is denoted $b$, the number of white balls just before a marked edge is
$w$.  

Let $\mu$ denote the probability that an edge has a marker on it.
Explain why
\begin{equation*}
  \mu = \frac{n}{N} = \frac{b}{B} = \frac{w}{W} \,.
\end{equation*}
\end{enumerate}
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