\magnification\magstep1 \nopagenumbers%\parskip=12 pt plus 4pt minus 4pt  
%\advance \hsize by .4in\advance \hoffset by -.2in
%\advance \vsize by .4in \advance \voffset by -.2in
\nopagenumbers
\def \chapno{5} %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. May 4}\smallskip\hrule\bigskip
\problem{ Phonon specific heat:}  The vibrational modes in a solid may be
considered as nonconserved bosons, whose energy $\epsilon(k) = \hbar c |k|$,
where $c$ is the speed of sound.  (This is a reasonable approximation for the
phonons in typical solids below a few tens of degrees Kelvin.)  Find the
specific heat at constant volume $C_v$ of a gas of such phonons in $d$
dimensions.  How does it vary with the temperature $\beta^{-1}$?  

\problem{ basis of Feynman Hellman theorem:} (cf. Chandler, p. 106) The energy
$E_c$ of an interacting system in configuration $c$ may be regarded as composed
of $E_{c0} + \lambda U_c$,  where $E_{c0}$ is the configuration energy of some
simplified system, \eg without interactions, and
$U_c$ is the interaction energy.  The parameter $\lambda$ is thus 1 for the real
interacting system.  
\item{a)} Show that for any $\lambda$ between 0 and 1 the
expectation value $\expectation{U}_\lambda$ may be related to a derivative of the
thermodynamic potential $\Omega$ with respect to $\lambda$.  
\item{b)} Using the result
of a), show how one may obtain the potential $\Omega$ of the interacting system
knowing only the $\Omega_0$ of the simple system and $\expectation{U}_\lambda$
for $0<\lambda<1$.

\problem{Fluctuations with constraint:} (cf. Chandler Problem 3.23).  A dilute gas consists
of $N$ molecules that may be in one of two states, A or B (\eg a four-carbon chain might
be in a boat-like $cis$ configuration or a zigzag $trans$ configuration).  
At a given moment there are 
$N_A$ molecules in state A and $N_B$ molecules in state B.  The free energies of
A and of B may not be equal, so that $\expectation{N_A}$ may be different from
$\expectation{N_B}$.
\item{a)} Find the variance of $N_A$, \viz
$\expectation{(N_A - \expectation{N_A})^2}$, in terms of $\expectation{N_A}$,
$\expectation{N_B}$ and $N$.

\problem{Fermi gas thermodynamics} (cf. Chandler, problem 4.18) It is often convenient
to express the thermodynamic potential as an expansion in the {\it activity} 
$z\definedas e^{\beta \mu}$.  
\item{a)} Verify that $f_{5/2}$ defined by the integral in Problem 4.18 can be 
expressed by the given power series: 
$f_{5/2}(z) = -\sum_{\ell=1}^\infinity (-z)^\ell/\ell^{5/2}$.  Are the numerical
prefactors correct?  What is the radius of convergence of this series? 
($f_s(-1)$ is known  as the Riemann zeta function $\zeta(s)$.)
\item{b)} Find the pressure in terms of $f_{5/2}$.
\item{c)} Find the average density in terms of $f_{3/2}$.
\item{d)} Show that the pressure is proportional to the average energy 
$\expectation{E}$ with a proportionality factor independent of $z$.  Find this factor.

\problem{Equation of state of fermi gas} (cf. Chandler, problems 4.19 and 4.20).
Find the dependence of the pressure on the density $\rho \definedas \expectation{N}/V$
for a fermi gas at very high or at very low temperature as given in these two
problems.