Computing the 3D Radial Distribution Function: An Advanced Analytic Approach

The radial distribution function, g(r), is ubiquitously used to analyze the internal structure of particulate systems. Applications range from molecular dynamic simulations to confocal microscopy of colloids. Measured particle coordinates are always confined in a finite sample volume. Computing g(r) is challenging once the radial distance, r, extends beyond the sample boundaries in at least one dimension. State of the art algorithms for g(r) use artificial periodic boundary conditions to circumvent this challenge. Ignoring the finite nature of the sample volume distorts g(r) significantly. Here, we present a simple, analytic algorithm for the computation of g(r) in finite samples. No additional assumptions about the sample are required. The key idea is to use an analytic solution for the intersection volume between a spherical shell and the sample volume. In addition, we discovered a natural upper bound for the radial distance that only depends on sample size and shape.