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The Ornstein Zernike equation
(Proc. Acad. Sci. Amsterdam, 17, 793 (1914))
Define the function h(r) to be
h r   g r  1
(1)
The function g(r) is the two-body correlation
function, the probability of finding a particle a
distance r from another particle at 0 defined in gofr.doc
and gofr2.doc. This probability in a liquid becomes
one for large enough r. The Ornstein Zernike
equation defines the direct correlation function c(r)
h(r )  c(r )    c(r ')h(| r  r ' |)d 3 r ' (2)
This says that the probability of finding a second
particle at a distance r from one at the origin differs
from one by the direct correlation due to the particle
at the origin plus the integral of the direct
correlation from all of the other particles. The value
of (2) comes from the fact that it relates a short
range function in r to a long range function in r.
Names such as Percus Yevick and Hyper-NettedChain or BBKGY are associated with
approximations to c(r). A technique due in part at
least to Hanson and Levesque is to define c(r) by
h(r)=hMC(r) r<R/2
c(r)=0
r>R/2
The integral in (2) is a 3-d convoution
integral. Direct evaluation of this is detailed in
Convolution in 3d.doc. Usually c and h are
functions of |r| which reduces the integrals to one
dimensional integrals as detailed in Sine
transform.doc. In transform space ConvDetail.doc
(3)
H  f   C  f   C  f  H  f 
or
Hf 
C f  
(4)
1 H  f 
or
C f 
Hf 
(5)
1  C  f 
Monte Carlo
Assume that as a result of the Monte Carlo
calculation described in gofr2.doc, the values of g(r)
or equivalently hMC(r) are known for vales of ri i =
1 to M. Define
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M
 2    hMC  ri   h  ri , c   (6)
2
i 1
where c is the set of c values at each point ri. Since
there are M constants and M points in the sum, it
should be possible to make 2 become zero. This
will not always result in a reasonable function c(r).
Three possible approaches are detailed below.
1.
A simple way to minimize 2 is to make it
the ftbm of the AMOEBA.
2.
Extremal.htm discusses non-linear
minimization of 2. In general this useses
nlfit\Welcome.htm. It requires inputting the
M values of hMC(ri) as data and the writing
of a routine Poly which calculates h(ri,c).
To efficiently calculate these values using
Sine transform.doc, they need to all be
calculated on a first call to Poly, stored and
then used later on subsequent calls. The
routine Poly also needs the partial of h with
respect to c(ri). This can be done
numerically, but the code is greatly
simplified if these derivatives can also be
calculated on a first call and stored
(Derivatives of h.doc).
3.
If the sum in 2 is extended to all N values
of r between r1 = 0 and r1 = N where h is
assumed small, and a wi is defined such that
wi = 1 for i < M and wi = 0 for i > M, then
(6) becomes similar to that in
WeightedLeast
SquaresbyIterationLambda.htm .doc. The
difference is that in the fit discussed there, H
is fitted rather than C. A version of this
designed for this fit is WeightedLeast
SquaresbyIterationLambda.doc. Each step
in this method produces a new estimate for
c(r). These can be combined using the
technique discussed in Mixing.htm .doc
A single constant c.
It is not necessary to set up a Monte Carlo
integration to deduce some things. In particular one
can assume that for a system of particles interacting
by means of a Lennard Jones Potential that there is
no chance of finding two particles close (at r = 0)
and that this determines most of the two body
correlation function. Assume that c(r) =  for r < r0
and is zero elsewhere. Then determine the single
constant  by requiring h(0) = -1.
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C  f ; , r0  
C  f ; , r0  
3.
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2
 2 
2
 2 
2
 2 
C  f  0; , r0  
5.
f
2
2 fr
0
 sin x  x cos x 0
3
f3
sin  2 fr0   2 fr0 cos  2 fr
In the limit that f0
C  f ; , r0  
4.
2
2
3

2 fr0 

 2 fr
 2 fr0 
 2 fr0 
3
6
2
f 
4  r0 
3
3
This is the
volume of the short range function c(r).
C  f ; , r0 
H  f ; , r0  
1   C  f ; , r0 
Sine transform.doc derives the fact that

2
h(r ; , r0 )   fH ( f ; , r0 )sin  2 fr df .
r0
In the limit that r0 this becomes

h(0; , r0 )  4  f 2 H ( f ; , r0 )df
0
Figure 1 Expected values of h(r) and c(r) in the single
constant (alpha) model.
The integral of g(r) over a box of radius R, where
4R3/3 =  is equal to the volume.
This is not a convergent integral, H(f) for large f goes
as cos(2fr0)/(f2) so that this integral becomes
6.
R
4  r 2  h(r )  1dr  
(7)
0
So that
R
4  r 2 h  r dr  0
(8)
7.
0
1.
2.
Set c(r) =  for r<r0
Sine transform.doc derives the fact that

2
C ( f )   rc(r )sin  2 fr dr (9)
f 0
r
2 0
C  f ; , r0  
r sin  2 fr  dr . Change
f 0
to standard form by letting x=2fr
2 fr
0
2
C  f ; , r0  
x sin  x  dx .
2
 2  f 3 0
Then the integral is

F
F
0
h(0; , r0 )  4  cos  2 fr0 df  4  f 2 H  f ; , r0  df
. Note that the non-convergent part is small and
oscillatory. There is some advantage to making Fr0
such that the sin(2Fr0)=0, but in general, this is
simply the abiguity of a sharp cutoff in r.
Define
fi  iF / N ;  f  F / N
N
fun( x)  4 f  fi 2 H  fi ; , r0   1
Then
i 1
8.
select a reasonable F and r0 and solve this for  using
Bracketing.htm .doc.
Finally now that  has been found and F and ro
assumed. The two body correlation function h(r) is
found by making an FFT of H using code contained
in Sine transform.doc
Simulated Monte-Carlo 2
At low densities, we expect g(r) to be
approximately 1 until the potential becomes
repulsive. Then we expect g(r) to be exp(VLJ(r)/(kT)). Thus we have as our first guess
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exp  VLJ  r    1
r 
hˆ1 (r ) 
0
r 
Transform this using Sine transform.doc
R
2
Hˆ 1 ( f )   rhˆ1 (r )sin  2 fr dr
f 0
Form
Cˆ1  f  
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Monte-Carlo
It is only slightly harder than the above to
use the Markov chain with a box of side L, to find a
realistic estimate of g(r) for r<r0 gofr2.doc
..\integration\MonteCarlo\gofr2.htm
R
Hˆ 1  f 
1   Hˆ  f 
+
+
1
+
+
R
+ +

cˆ1 (r ) 
2
fCˆ1 ( f )sin  2 fr df

r0
using Monte Carlo techniques. Obviously there is
an enormous premium for making small N give the
large N result. The close in parts for r<R/2 rather
rapidly reach the same result that one would find in
an infinite system, while the parts for r>R are
obviously very biased by the fact that they are
averages over periodic repeats.
This is only one of many possible ways to define
c(r), others have
Set I =1
cˆI  r  r  
0 r 
1.
cI  r  
2.
CI ( f ) 
3.
HI  f  
R
2
rcI (r )sin  2 fr dr
f 0
CI  f 
1   CI  f 

2
fH I ( f )sin  2 fr df
r 0
4.
hI (r ) 
5.
  4  r 2 hˆ1  r   hI  r  dr


2
I

2
0
6.
If 2 exit
Set I=I+1
7.
Form hˆI (r ) 
exp  VLJ  r    1
hI 1  r 
r 
r 
R
8.
9.
2
Hˆ I ( f )   rhˆI (r )sin  2 fr dr
f 0
Hˆ I  f 
Cˆ I  f  
1   Hˆ I  f 

10.
11.
cˆI (r ) 
2
fCˆ I ( f )sin  2 fr df

r0
Goto 1
The above procedure may not converge. There are,
however, M values of h for r <  and M values of c,
so in principle it is possible to form these.
As can be seen to the right the 5 values of h
computed by Monte-Carlo are the result of 5
integrals over all r’ of the function c(r’)h(r-r’) with r
taken as each of the 5 values shown. This gives 5
equations for the 5 desired values of c(r) which in
principle could be solved by minimizing
5

   h  ri    d r ' c(c , r ')h  ri  r ' 
2
i 1
3

2
(10)
where the constant vector c in c can be taken simply
as the values of c at the 5 points where it is assumed
to be non-zero. The integral uses h values between
R/2 and R which must be calculated from
h(r )    c(c , r ')h(| r  r ' |)d 3 r ' (11)
for each assumed value of c. Note that we have
introduced a “fictitious” short ranged function c(r)
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which produces the long range function h(r) and in
addition given a scheme for calculating this
function. In essence the Ornstein Zernicke equation
is fundamental in liquid theory and many people
without adequate thought about Fourier transforms
have tried to find efficient ways to do the 3
dimensional integral for all 3d values of r
amounting to a 6 dimensional set of points to
calculate.
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