# Pair Distribution Function

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 $r=0$. In the absence of spatial correlations, which only applys for a classical gas composed of non-interacting particles, the function $g(r)$ is identically equal to unity. For real system, $g(r)$ is generally different from unity. It is then natural to define a function $v(r)$

$v(r) = g(r) -1$,

as a measure of the degree of spatial correlation in the system. $v(r)$ is generally referred to as the pair correlation function. In the absence of spatial correlations, $v(r)$ is zero.

## Reference

[1] R. K. Pathria, "Statistical Mechanics", 2nd ed., Butterworth-Heinemann, 1996