Radial Distribution Function

The goal of this section is to introduce the radial distribution function$g({\bf r})$ (also known as pair correlation function), a central quantity in studies of fluids since it determines the average density of particles at a coordinate ${\bf r}$ relative to any particle in the fluid.

The radial distribution function is defined as follows

$$g({\bf r}) = \rho^{2/N}(0, {\bf r})/ \rho^2,$$ (388)

where $\rho=N/V$ is the density of a fluid of N particles in a container of volume V and $\rho^{2/N}(0, {\bf r})$ is the probability that a particle is at r when there is another particle at the origen of coordinates.

The probability $P^{2/N}({\bf R}_1,{\bf R}_2)$ that particle 1 is found at ${\bf R}_1$ when particle 2 is at ${\bf R}_2$, in an $N$ particle system, is

$$P^{2/N}({\bf R}_1,{\bf R}_2) = \frac{\int dr^N \delta({\bf r}_1-{......= \frac{\int dr^{N-2} e^{-\beta V(R_1,R_2,r_3, ...r_N)}}{\int dr^N e^{V(r^N)}},$$ (389)

and the probability $\rho^{2/N}({\bf R}_1,{\bf R}_2)$ that a particle (i.e., any particle) is found at ${\bf R}_1$ when another one (i.e., any other one) is at ${\bf R}_2$ is

$$\rho^{2/N}({\bf R}_1,{\bf R}_2) = \frac{N!}{(N-2)!} P^{2/N}({\bf R}_1,{\bf R}_2)= N (N-1) P^{2/N}({\bf R}_1,{\bf R}_2).$$ (390)

In particular, in a fluid where the interaction between particles can be neglected,

$$P^{2/N}({\bf R}_1,{\bf R}_2) = \frac{1}{V^2},$$ (391)

or

$$\rho^{2/N}({\bf R}_1,{\bf R}_2) = \frac{N (N-1)}{V^2} \approx \rho^2,$$ (392)

and $g({\bf r})=1$. In an atomic fluid (e.g., liquid argon), $g({\bf r}_1,{\bf r}_2)=g(r)$, where $r=\mid {\bf r}_1 - {\bf r}_2 \mid$, since the fluid is uniform and isotropic. The presence of an atom at the origen of coordinates excludes other particles from all distances smaller than the radius of the first coordination shell where $g(r)$ has a maximum (see figure).

The presence of the first coordination shell tends to exclude particles that are closer than the radius of the second coordination shell, where $g(r)$ has another maximum. This oscillatory form for $g(r)$ persists until $r$ is larger than the range of correlations between the particles. At distances larger than the correlation length $g(r)=1$, since $\rho^{2/N}(0,r) \approx \rho^2$ for uncorrelated particles. Therefore, $h(r)=g(r)-1$ describes the deviation from the asymptotic limit of uncorrelated particles (i.e., an ideal gas) and the product $\rho g(r)$ describes the average density of particles at a distance $r$ from any particle in the fluid.

Note that $\rho g(r) 4 \pi r^2 dr$ is the number of particles at a distance between r and r+dr from any particle in the fluid. Therefore, the calculation of $g(r)$ involves averaging the number of particles at a distance r from any particle in the system and dividing that number by the element of volume $4 \pi r^2 dr$.

Exercise (due 04/24/03):

Compute the radial distribution function g(r) for a fluid of argon atoms at a constant T,N,V using the program developed in the assignment of simulating annealing.