Deterministic reaction models with power-law forces

TL;DR

This paper analyzes a one-dimensional particle system with power-law attractive forces, developing an exact mean-field Fokker-Planck approach to understand gap distributions and the impact of correlations, including in stationary states with random inputs.

Contribution

It introduces an exact mean-field Fokker-Planck framework for power-law attraction models and explores the role of correlations in gap distributions.

Findings

Exact solution for the mean-field gap distribution.

Correlations significantly affect the distribution shape.

Random input leads to a stationary state with reduced correlation effects.

Abstract

We study a one-dimensional particles system, in the overdamped limit, where nearest particles attract with a force inversely proportional to a power of their distance and coalesce upon encounter. The detailed shape of the distribution function for the gap between neighbouring particles serves to discriminate between different laws of attraction. We develop an exact Fokker-Planck approach for the infinite hierarchy of distribution functions for multiple adjacent gaps and solve it exactly, at the mean-field level, where correlations are ignored. The crucial role of correlations and their effect on the gap distribution function is explored both numerically and analytically. Finally, we analyse a random input of particles, which results in a stationary state where the effect of correlations is largely diminished.