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\line{Physics 352 \hfill Spring, 1995 \hfill Problem Set \chapno \hfill Due
date: 
Th. May 25}\smallskip\hrule\bigskip
\def\Tr{{\rm Tr~}}
\problem{Transfer-matrix correlations} The transfer-matrix method of the last
problem set can be used to get spatial information about a system.  It is useful
to define the matrix $\hat n\definedas \left (\matrix{0& 0\cr 0& 1}\right )$
corresponding to the occupation number at a site (I may have the 1 in the
wrong corner).  As in the last problem set the Boltzmann weight $f_c =
\exp[-\beta v
\sum_{\expectation{i,j}} n_i n_j +
\beta \mu \sum_i n_i]$, and the lattice has $L$ sites.  
\item{a)} Show that $\expectation{n_0 n_i} \definedas \sum_c n_0 n_i f_c /
\sum_c f_c = \Tr \hat n ~\hat q^i~ \hat n \hat q^{L-i}/\Tr \hat q^L$.  Find the
limiting form as $L \goesto \infinity$.

\item{b)} Using the eigenvalues and eigenvectors of $\hat q$ that you found in
the last problem set, calculate $\expectation{n_0 n_i}$.  

\item{c)} For large $i$, the $\expectation{n_0 n_i}$ decays exponentially to
a constant: $\expectation{n_0 n_i} = C_1 + C_2 e^{-i/\xi}$.  Calculate
$\expectation{n}$ and express $C_1$ in terms of it.  What is the correlation
length
$\xi$ in terms of $\beta v$ and $\beta \mu$?

\problem{Ornstein-Zernike on 1d lattice}
\item{a)} Use Handout E to find the Ornstein-Zernike susceptibility for the
one-dimensional lattice gas of the previous problem.
\item{b)} Find the correlation length and compare with the previous problem.

\problem{Monte-Carlo correlations} Using your Monte-Carlo simulation from the
last problem set, you can measure $\expectation{n_0 n_i}$.  It is just the
average of $n_k n_{k+i}$ over all sites $k$ and over many monte-carlo samples.
\item{a)} Take $\mu=0$ and choose $\beta v$ so that the correlation length
$\xi$ as calculated above should be be 5.  Do a monte-carlo simulation on your
32-site lattice from last time.  Measure the  $\expectation{n_0 n_i}$ for all
$i$.   (You may accumulate your samples of $n_k n_{k+i}$ for all $i$ at once. 
You need not sample on every time step.  Just sample after the system has had a
chance to change substantially, \viz every 100 time steps).  
\item{b)} Make a semi-log plot of $\expectation{n_0 n_i}$ \vs $i$, and compare
the decay length with the predicted correlation length of 5.  
\bye