The two body correlation function
Definition
We want a function g(r1,r2) that becomes 1 for large values of the argument
Define the two-body correlation function to be
This means that the integral
 2  d 3r1  d 3r2 g  r1 , r2   N  N  1

(1)

g  r1 , r2  
2  d 3..d N 2 (r1 , r2 ,..., rN )
(2)
2
 d1...d N (r1, r2 ,..., rN )
Define the single variable g to be the average of (2) over r2 for r1 = r + r2
g r  
1
g  r  r2 , r2  d 3r2
 

(3)
Define
1
g r  
4
1
2
1
0
 d   g  r ,  ,   d
(4)
Note that with
4
   R3
3
(5)
That
R
4  r 2 g  r  dr  
(6)
0
The expectation value of the potential is defined to be
1 N

d

...
d

v  ri  rj   2 (r1 , r2 ,..., rN )
 1 N 2  
i i

V 
2
 d1...d N (r1, r2 ,..., rN )
(7)
All coordinates are equivalent in this definition leading to
V 
2
2
N  N  1  v  r1  r2  d1d 2  d 3..d N (r1, r2 ,..., rN )
2
 d1...d N (r1, r2 ,..., rN )
22
Or with an error of 1/N
V 
1 2
 d 1d 2v  r1  r2  g  r1 , r2 
2 
(9)
Let r = r1-r2 in the r1 integration
V 
2
2
 v  r  d r  g  r  r , r  d r
3
3
2

2
2
(10)

Substitute in the value for g(r) from (3)
V 
N
v  r g  r  d 3r

2 
Change to spherical coordinates
(11)
(8)
V
N
4 2
r v  r  g  r  dr
2 0
R

(12)
Note that V is going as N2 so that this <V> is still large. This is because the g includes an N-1 from the fact
that any of N-1 other particles could be next to particle 1.
The two body correlations function is crudely related to the scattering function in IbachLuth.doc .htm. A
prescription for finding g(r) using Monte Carlo methods is described in gofr2.doc.htm Contains HW 22 –
an interesting exercise of medium difficulty.