# Pair Distribution Function

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For a system with N particles, the configurational distribution function, $F_N(r_1,...,r_N)$, of the system appears in the nature of a probability density and satisfies the normalizaiton condition

$\int F_N(r_1,...,r_N) d^3r_1 ... d^3r_N =1$.

Integrating $F_N(r_1,...,r_N)$ over the coordinates $r_2$, ..., $r_N$ and multiplying the result by N, a single particle distribution function can be obtained

$F_1(r_1)= N \int F_N(r_1,...,r_N) d^3r_2 ... d^3r_N =n(r_1)$.


The function $F_1(r_1)$ represents the particle density at the point $r_1$. The two-particle distribution function is now defined as

$F_2(r_1,r_2)= N (N-1) \int F_N(r_1,...,r_N) d^3r_3 ... d^3r_N =n^2 g(r)$,

where $r=r_2-r_1$. The above equation defines the pair distribution funciton $g(r)$ of the system. The product $g(r) d^3r$ determines the probability of finding a particle in the volume element $d^3r$ around the point $r$ when there has been a particle at the point $r=0$