Two body correlation function

Approximate the two body correlation function as

Eqn 1

Where S_k(r) is as shown in figure 1

Figure 1 S_K(r)

The two-body correlation function is defined to be

Eqn 2

Form

Eqn 3

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Minimize this by setting the partials with respect to c_m equal to zero.

Eqn 4

The array of k,m S values becomes

Eqn 5

Thus the equation for the coefficients becomes

Eqn 6

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Any coordinate can be 1 and any other can be 2 so that this is

Note that this merely amounts to counting the number of distances in the ranges r_m to r_m+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.
Evaluate the change in . If . Note that this will not happen in the beginning if all particles start on lattice sites. If with probability

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 ~ 1000⁰ 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.