\magnification\magstep1 \nopagenumbers%\parskip=12 pt plus 4pt minus 4pt  
\def \chapno{1} %roman for chapters; arabic for problem sets; 
\newcount\notenumber \notenumber=1
\def\problem#1{\par\medskip\noindent \chapno .\the\notenumber \global\advance
\notenumber by 1 ~{\sl #1}\quad}
\def\?#1{ } 
\line{Physics 352 \hfill Spring, 1995 \hfill Problem Set \chapno \hfill Due
date: 
Th. April 6}\smallskip\hrule\bigskip
\def \RND {{\tt RND}}
\problem{Computer warmup: random numbers} Most computers provide a ``random
number generator" called, for example \RND, or {\tt random(1)}, that
is supposed to produce uniformly-distributed random numbers between 0 and
1.  Random numbers are used extensively in monte-carlo simulations.  This
exercise asks you to perform some simple tests on the random number generator
you intend to use for your simulations later in the course.  a) Call the random
number generator a million times (or 100,000 times if your computer is slow) and
note the elapsed clock time.  How many random numbers can you generate per
second?  b) Generate a million random numbers between 0 and 100 (by
multiplying \RND by 100).  Count the number of occurences between 0 and 1,
between 1 and 2, 2 and 3, ...99 and 100.  (You could define an integer array
$M(J)$ of dimension 100 and if your random number falls between $J-1$ and $J$
you would increment
$M(J)$: $M(J) = M(J) + 1$.)  Verify that all the $M(J)$ are about the same. 
That is, find the relative variance of the $M$'s $\definedas \expectation{(M -
\expectation{M})^2}/\expectation{M}^2$ and verify that this relative variance is
about
$1/
\expectation{M}$.  
c) Using your \RND, create a random number generator that makes a ``circular"
distribution.  That means, the probability $p(x)$ for a number $x$ is $const.
\sqrt{1-x^2}$ for $x$ between 0 and 1.  You can create this generator by defining
a suitable function
$g($\RND$)$.  Hint: Note that the probability $P_0(y)$ that \RND $<y$ is simply
$y$.  For some arbitrary increasing function $g$, what is the probability $P(z)$
that the number
$g($\RND$)<z$?  How is this $P(z)$ related to the desired $p(x)$?
\omit{$y = P(y) = const \integral_0^z dx\sqrt{1- x^2}
= $}
\def\dslash{{\slash\mkern-9mud}}
\problem{Rubber band entropy} Chandler's exercise 1.2 gives two candidate
expressions for the thermodynamic entropy of a rubber band of length $L_0$ with
a fixed amount of rubber per unit length $n/L_0$.  If the rubber band is
stretched to a different length $L$, work is done.  The work $\dslash W$ has
the form $f dL$, where the tension $f$ depends on the $L$, $L_0$, the
temperature $T$ and material constants $\gamma$, $\theta$ and amount of
rubber (mole number) $n$.  a) Which of the two candidate formulas for entropy is
properly additive for arbitrary amounts  of rubber band $n$?  b) Using the
relations between thermodynamic entropy and work, deduce the force $f$ for a
given length
$L$.  c) Exercise 1.4.  Given two stretched pieces of  rubber band with different
temperatures $T$, lengths $L$ and mole numbers $n$, determine their final
energies and temperatures if they are put in contact.  You may assume that
the two pieces are cut from the same rubber band, so that they both have the
same $\ell_0 \definedas L_0/n$.  

\problem{Extremal property of alternative thermodynamic functions}  Cf.
Exercise 1.7.
 The text defines the Gibbs free energy $G$ and states that it is a ``natural"
thermodynamic function of temperature $T$ , pressure $p$ and number of particles
(or moles)
$n$.  Suppose that a system at fixed $T, P, n$ is further constrained by
specifying a variable $Y$, whose equilibrium value is $Y_0$.  
$\bullet$ We suppose that upon the releasing the constraint $Y$ increases in
value 
[I am trying to get you to prove that the $\delta G$ defined in the text
is non-negative.  I goofed in rewording the problem.  The preceding addition
marked by $\bullet$ should fix this.  Sorry. T.W.].
 a) Show from the
Second Law that the change in
$\Delta G \definedas G(T, p, n, Y) - G(T, p, n, Y_0)$ must be positive. 
 b) Show that the derivitive $\left .\partial G/\partial Y \right |_{Y=Y_0}$
must be non-negative.