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\Large \bf
Computational Physics \\
\large 
\vspace{0.2cm}
29-January-2001\\
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\noindent 
{\bf Ex.~1a: Random Walk  on a square lattice} 

In a random walk (RW) on a two-dimensional square lattice, at each step,  
a particle can move up, down, left or 
right, with equal probability. Make use of a random number generator to 
decide the direction of each step. In fortran 90 you can use the built in 
function RANDOM\_NUMBER($ran$) which returns values of $ran$ uniformely 
distributed between 0 an 1 ($0 \le ran <1$). 
You must initialize (or restart) the random sequence by calling 
the function RANDOM\_SEED.
An example can be found in the section fortran90.

Make some test of the 
randomness of this generator, either before you start or during the calculation,
for instance by checking how many steps have been made to the right, left, up
or down. We are not interested in keeping track of the actual trajectory 
in space but are only interested in calculating
the mean square displacement  as a 
function of the number of steps $nstep$ for $nstep=2,4,8,16,32,64$
with an accuracy of 5~\% or less. 

Establish the dependence of the mean square displacement 
\begin{equation}
<\Delta R^2(N)>=<x^2(N)>-<x(N)>^2+ <y^2(N)>-<y(N)>^2
\end{equation}
as a function of the number of steps
$nstep$ by assuming the asymptotic $N$ dependence  of $<\Delta R^2(N)>$ to be
\begin{equation}
<\Delta R^2(N)> \sim N^{2 \nu}  ~~~~~~~~~~~  (N>>1)
\end{equation}

If $\nu \sim 1/2$, estimate the self-diffusion coefficient $D$ given by 
\begin{equation}
<\Delta R^2(N)> \sim 2dDN
\end{equation}

with $d$=2 in two dimensions.

A useful exercise to test your program is to enumerate all the 
possible walks on a square lattice of size $N \times N$, with $N=2,4$ and 
obtain exact  results for $<x(N)>$, $<y(N)>$ and  $<\Delta R^2(N)>$ 
to compare with your numerical results. 
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