I'm trying to understand the definition of the n-th order correlation function. My aim is to translate the math into a numerical implementation in order to compute the correlation function g(n) for a distribution of spherical particles with n={3,4,5,…}.
I'm going to present you what I have understood for now, in order for you to get an better insight of what I'm looking for. (To reduce the verbiage of the following, I over-abbreviated sometimes r1,r2 into r, sorry if it kills the readability)
I'm used to the radial distribution (or pair distribution function) that we call g(r) (which should be called rigorously g(2)(r)). I want to derive the expression of g(2)(r) from the definition of g(n) (because I hope that if I'm able to do it for n=2, I will be able to do it for any n !)
So let's start with the definition of g(n) one can find in wikipedia or any stat mech book :
ρ(n)(r1…,rn)=ρng(n)(r1…,rn),
with
ρ the particle number density and
ρ(n) the probability of (all the permutations of) elementary configuration
(r1…,rn) :
ρ(n)(r1…,rn)=N!(N−n)!1ZN∫⋯∫e−βUNdrn+1⋯drN.
with
UN(r1…,rN) the potential energy of the configuration and
ZN the configurational integral, taken over all possible combinations of particle positions.
Rewriting the previous with n=2 leads to :
ρ(2)(r1r2)=N(N−1)1ZN∫⋯∫e−βUNdr3⋯drN.
From that starting point, I should be able to derive the expression for g(2)(r1r2) but I have no clue about the origin of the Dirac Delta function that appears in the definition of g(2)(r1r2) or ρ(2)(r1r2)
ρ(2)(r1r2)=⟨∑i∑j′δ(r1−ri)δ(r2−rj)⟩
I think there is something appearing from the potential energy that was defined before, but I'm not able to understand exactly the origin.
Any help will be highly appreciated :) Thank you in advance !
(Note : related questions like "Use and understanding of higher-order correlation functions" is absolutely not helpful, and the reference on the wikipedia page leads to a paper that has nothing to deal with correlations functions IM(H)O...)
This post imported from StackExchange Physics at 2014-06-08 08:17 (UCT), posted by SE-user Pascail