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\line{Physics 352 \hfill Spring, 1995 \hfill Problem Set \chapno \hfill Due
date: 
Th. April 21}\smallskip\hrule\bigskip

\problem{Ergodic Hypothesis tested}  A particle is confined to a square 
lattice of integers $(i, j)$ bounded by an irregular boundary: $i_{min}(j)< i < 
i_{max} (j)$.  The sequence of  $i_{min}$'s and  $i_{max}$'s is 
{\tt\obeylines
1 ,1 ,1 ,1 ,1 ,1 ,1 ,2, 3, 4, 5, 6, 7, 8, 8, 9 ,10, 9, 8, 7, 7, 6
5 ,5 ,4 ,3, 4, 5, 6, 7, 8, 9 ,10,10,10,10,11,12,12,13,14,15,16, 17
}
\noindent
The lower boundary is $j=1$; the upper boundary is $j=22$.  Thus the 
configurations $c$ of this system are the points $(i, j)$ lying within the 
boundary.
The object is to investigate how an ergodic process reaches all allowed 
configurations with equal probability.   The ergodic process to be used is a 
smearing process of the type studied in Problem 2.3.  It is a random walk, a
probabilistic procedure for picking a configuration $c'$ based on the current
configuration $c$.
The procedure is to pick a point $(i', j')$ adjacent to $(i,j)$ at random.  
If this new point contacts the boundary, the point is restored to $(i, j)$.
Otherwise, the new configuration is the new point $(i', j')$.

\item{a)} Show that the conditional probability $q(i, j, i', j')$ for moving 
from $(i, j)$ to $(i', j')$ obeys detailed balance: 
$q(i, j, i', j') = q(i', j', i, j)$.  What is the value of $q$ for a 
nearest-neighbor step?  In view of the revised Problem 2.3, the process must 
increase the entropy of the probability distribution $p(i, j)$, unless it leaves
$p(i, j)$ unchanged.  

\item{b)} By using the language of Problem 2.3, show that the process 
changes the distribution $p(i, j)$ if any two neighboring sites have unequal 
$p$.  This means that as long as any $p$'s remain unequal the entropy must 
continue to increase.  Thus by iterating the random walk the $p(i, j)$ must 
approach equal values, independent of the initial state.

\item{c)} Implement the random walk described above as a computer program.  Sketch
the allowed region.  
Define a matrix $M(i, j)$ and increment $M(i, j)$ by 1 whenever the site $(i, j)$
is  visited.  Find the minimum, maximum, average $\expectation{M}$ and standard 
deviation 
$\sigma \definedas \sqrt{\expectation{M^2} - 
\expectation{M}^2}/\expectation{M}$.  Do this after 1000, 2000, 4000, 8000, 
... steps of the walk.  If you can make a density plot showing how $M$ 
develops in time, it would be nice.  Try a few different starting points.  
How does $\sigma\aboutequal \expectation{M}^{-1/2}$ vary with the number of
iterations $t$?  Show by a plot that $\sigma \goesto (t/t_0)^{-1/2}$.  What
is the significance of the parameter $t_0$?
Compare with your result from the random 
number problem in Problem set 1?  A sample program that does this appears
in the course bulletin board as {\tt ranwalk.bas}.

\problem{ Equipartition of energy}  Many simple systems have energies which are
the sum of {\it squares} of system variables.  Thus in an ideal gas, the energy is
given by 
$$
E=\sum_i \sum_{\alpha=1}^3 {1\over 2} m \left (v_i^{(\alpha)} \right )^2.
$$
  Here the $i$
index is over the $N$ particles in the system, and the $\alpha$ runs over the
components of the velocity $v$.  The energy is the sum of squares of $3N$
independent variables.  
\item{a)} Show that when the gas above is in equilibrium with a
reservoir at temperature $1/\beta$, the energy associated with each variable is
a fixed multiple of $1/\beta$; find this multiple.  
\item{b)} Generalize your result in
a) to the case $E=\sum_j E_j$, where $E_j = b_j x_j^2$, and the $x_j$ are
arbitrary independent variables of the system.  
\item{c)} OPTIONAL.  Generalize your
result further to the case of a general quadratic form, where $E = b_{ij} x_i
x_j$.  If the matrix $b$ has $N$ positive eigenvalues and no negative ones, find
the total energy $\expectation{E}$ when the system is at  temperature $1/\beta$. 

\problem{Limited equilibration} Given two spins in a
magnetic field with configurations  + and - and energies $\plusorminus
\epsilon$, in an initial distribution
$p_+, (1-p_+)$ and $q_+ , (1-q_+)$, find the final $p_+$ and $q_+$ if the two spins
are allowed to equilibrate with each other.