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\line{Physics 352 \hfill Spring, 1995 \hfill Problem Set \chapno \hfill Due
date: 
Th. May 18}\smallskip\hrule\bigskip

\problem{Transfer matrix for lattice gas (Cf. Chandler prob. 5.21)}  The lattice
gas is a very useful paradigm of an interacting system.  The one-dimensional
lattice gas has been used to understand, \eg the helix-coil transition of DNA
molecules, and the high-conductivity states of some one-dimensional conductors. 
This model consists of a ring of $L$ sites $i = 1...L$, each of which may be
either empty or occupied: $n_i = 0$ or 1.  
In addition to the chemical potential, 
There is an
interaction energy
$v$ for every pair of adjacent sites that are not the same.  Thus the energy
of the system $U = v\sum_i n_i(1-n_{i+1}) + n_{i+1}(1-n_i)$.  A positive $v$
means that two particles have a lower energy together than they do apart.  Thus
it amounts to an attraction. There is also a chemical potential $\mu$.
The partition function $Q$ is thus given by 
$$
Q = \sum_{n_1, ..., n_L = 0}^1 \exp[- \beta  v\sum_i n_i(1-n_{i+1}) +
n_{i+1}(1-n_i)  + \beta \mu \sum_i (n_i - \half) ]
$$
Since the lattice sites are in a ring, when $i=L$, $i+1 = 1$.  The $- \half$ term
can have no physical effect: I add it only to preserve the particle-hole
symmetry.

\item{a)} By expressing the exponential as a product and noting that a given
$n_i$ appears in only two terms of the product, show that $Q$ can be written as 
$Tr \hat q^L$, where
$$
\hat q = \left(\matrix{
e^{ \beta \mu/2} & e^{-\beta v} \cr
e^{-\beta v}  & e^{- \beta \mu/2}
}\right )
$$
\item{b)} Find the eigenvalues $\lambda_>$ and $\lambda_<$ of $\hat q$.
\item{c)} Express $Q$ in terms of these eigenvalues.
\item{d)} Noting that the smaller eigenvalue $\lambda_<$ contributes negligibly
to $Q$ as $L\goesto\infinity$, find the free energy $\beta p L = -T \log Q$.
\item{e)} Find the the susceptibility $\chi/\beta = 
{\partial \expectation{N}\over \partial \beta \mu} = \expectation{N^2 -
\expectation{N}^2}$ when $\mu=0$.  How does it increase for $\beta v >> 1$?

\problem{ Monte-Carlo lattice gas}  A simple way to simulate a one-dimensional
lattice gas is to pick a site at random and attempt to reverse its state (from
empty to occupied, or vice-versa).  If the attempt results in a reduced energy,
it is kept.  If it results in an increased energy, it is kept with a probability
$\exp(-\beta \Delta E)$; otherwise the site is returned to its original state. 
This is process is repeated many times with all $L$ sites.  
\item{a)} Using a 32-site
lattice, simulate the system of the previous problem, taking $v=1$ and
$\mu=0$.  How does
$\expectation{N^2 - \expectation{N}^2}$ 
increase with $\beta$?  Does it agree
with the analytical results of the previous problem?  Why does it fail for large
$\beta$?

\problem{Virial expansion} If the volume fraction $\phi\definedas
\expectation{N}/L$ in a lattice gas is very small, it must behave nearly like an
ideal gas with $ p L = -T \log Q = -T
\expectation{N}$.  As $\expectation{N}$ becomes larger, $p$ may not follow the
ideal gas law.  Using the results of the first problem above, expand $\log Q$
and $\expectation{N}$ to second order in the activity $z \definedas
e^{\beta\mu}$.  From this expansion find an expression $p(\phi)$ up
to order $\phi^2$.  How is the quadratic term related the
interaction energy $v$?  
\bye