\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{6} %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 11}\smallskip\hrule\bigskip

\problem{Monte-Carlo Chemical Equilibrium}
Two particles of type $A$ live on a one-dimensional lattice of length $L$.  The
configuration of the system consists of the two co-ordinates $(i, j)$ of the
particles.  The particles have an attractive interaction: if they are on the
same site, $i=j$, the energy is
$-E$.  Otherwise it is zero.  The system may be simulated by a Monte-Carlo
method.  Select a particle at random and move it to a random site of the
lattice.  If the move increases the energy (\ie breaks a dimer), it is rejected
with a probability $1 -e^{-E/T}$.  In equilibrium the two particles form a dimer
in some fraction $p$ of the configurations.  
\item{a)} Find the concentration $[A_2]= p/L$ of pairs and the concentration
$[A]=2(1-p)/L$ of single particles.  Verify that $[A_2]/[A]^2$ is independent
of $L$ and depends only on $E/T$.  Is the functional form consistent with the 
law of mass action (Chandler, Chapter 4)?  Consider $L= 16, 64, 128, 1028$ and
$E/T = .5, 1, 2$. 
\item{b)} Generalize to two species $A$ and $B$.  The energy is now $-E$  for
every $AB$ on the same site.  If any further particles should be on the same
site as a pair, let its energy be zero.  Simulate the case of two
$A$'s and two $B$'s.  The configuration now consists of $(i_1, i_2, j_1, j_2)$.
To avoid overcounting indistinguishable configurations we may require
$i_1\lessthanorequal i_2$ and $j_1
\lessthanorequal j_2$.  Verify the mass-action law
${[AB] \over [A][B]} = {\rm const}~ e^{E/T}$.  What is the constant?  In this
system the law is not exact for all
$L$.  How large is the error for $L = 16$?
\item{c)} Verify the law of mass action with one $A$ and three $B$'s in the
system.

\problem{Thermal wavelength $\lambda_{th}$} In class we found that the number
of particles $N$ in a sample with length $L$ in all $d$ dimensions is related to
the chemical potential
$\mu$ by 
$N = (L/\lambda_{th})^d e^{\mu/T}$, where the thermal wavelength $\lambda_{th}
= K(d) (mT/\hbar^2)^{1/2}$.  Here $m$ is the mass of a particle and $K(d)$ is a
numerical constant.  
\item{a)} What is the value of $K(d)$ in two and in three dimensions?
\item{b)} How is $\lambda_{th}$ related to Chandler's $\lambda_i$ defined on p.
112 and $\lambda$ defined in Prob. 4.18?

\problem{Langmuir Isotherm revisited}The Langmuir Isotherm of Problem 4.5 can
be viewed another way.  Consider a set of identical sites that may each be
occupied  by zero or one particle.  The ratio of particles to sites is the {\it
volume fraction} $\phi$.  Thus the probability that a given site is occupied is 
$\phi$.
\item{a)} Find the entropy per site directly in terms of volume fraction
$\phi$.  Since the entropy is proportional to the number of sites, it suffices
to consider the entropy of a single site.
\item{b)} At temperature $T$ find the chemical potential $\mu$ by expressing it
as a derivitive of
$S$.
\item{c)} Compare with the $\phi(\mu)$ you found in Problem 4.5.  Recall that 
the pressure $p$ of an ideal gas is given by $pV = T N = T \lambda_{th}^3
e^{\mu/T}$.
\bye