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Minimize this by setting the partials with respect to cm equal to
zero.
Two body correlation function
 2

cm
Approximate the two body correlation function as
M
g A  r1 , r2    ck S k  r1  r2
k 0

M

2   ck Sk  r1  r2   g  r1 , r2   S m  r1  r2  d1d 2  0
 k 0

Eqn 1
Where Sk(r) is as shown in figure 1
Eqn 4
The array of k,m S values becomes
ck  S k  r1  r2  S m  r1  r2  d1d 2 
ckV  S k  x  S m  x  d 3 x  ckV 4 km
rk 1
 r dr
2
rk
4
 ckV  km   rk31  rk3 
3
Eqn 5
Figure 1 SK(r)
Thus the equation for the coefficients becomes
The two-body correlation function is defined to be
4
cmV   rm31  rm3    S m  r1  r2  g  r1 , r2  d 1d 2
3
g  r1 , r2  
N   2  r1 , r2 , r3 ,
, rN  d 3
  r , r , r ,
, rN  d1
2
1
2
d N
3
d N

Eqn 2
N  S m  r1  r2   2  r1 , r2 , r3 ,
  r , r , r ,
2
1
Form
 2    g A  r1  r2   g  r1 , r2   d 1d 2
2
3
, rN  d 1d 2 d 3
, rN  d 1
d N
d N
2
Eqn 6
2
M

    ck S k  r1  r2   g  r1 , r2   d1d 2
 k 0

__________________________________________________
Eqn 3
_________________________________________________
Any coordinate can be 1 and any other can be 2 so that this is
4
cmV   rm31  rm3  
3
N


N
1
Sm ri  rj  2  r1 , r2 , r3 ,

2 N  N  1 i 1
, rN  d1d 2 d 3
d N
j i
  r , r , r ,
2
1
2
3
, rN  d1
d N
Note that this merely amounts to counting the number of distances in the ranges r m to rm+1 and dividing by the appropriate volume
factor.
The Markov chain method is to put N particles in the volume V. Move particle j a small amount  .
N

 
V   v ri  rj    v ri  rj
Evaluate the change in
i j
beginning if all particles start on lattice sites. If

. If
V  0 rj  r  
V  0 rj  r  
with probability
. Note that this will not happen in the
exp  V 
Assignment
Use the Markov chain – see ..\..\integration\MonteCarlo\Expectation value of H.htm for details -with a periodically repeated
box containing ~64 atoms. Combine this with the OZ equations ..\..\Fourier\OZeqn.doc .htm to find the two body correlation
function for a Lennard Jones Liquid. Do this at both a high T ~ 1000 0 and for a low T ~ 100. Send me plots of the various g(r) ‘s.
Periodic distances are discussed in ..\..\integration\MonteCarlo\Periodic Distance.htm.