Pair Distribution Function

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

<math>\int F_N(r_1,...,r_N) d^3r_1 ... d^3r_N =1</math>.

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

<math>F_1(r_1)= N \int F_N(r_1,...,r_N) d^3r_2 ... d^3r_N =n(r_1)</math>.

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

<math>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)</math>,

where <math>r=r_2-r_1</math>. The above equation defines the pair distribution funciton <math>g(r)</math> of the system. The product <math>g(r) d^3r</math> determines the probability of finding a particle in the volume element <math>d^3r</math> around the point <math>r</math> when there has been a particle at the point <math>r=0</math>. In the absence of spatial correlations, which only applys for a classical gas composed of non-interacting particles, the function <math>g(r)</math> is identically equal to unity. For real system, <math>g(r)</math> is generally different from unity. It is then natural to define a function <math>v(r)</math>

<math>v(r) = g(r) -1</math>,

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