Master equation for the probability distribution functions of forces in soft particle packings

TL;DR

This paper develops a master equation framework for the evolution of force distributions in soft particle packings, incorporating contact and gap dynamics, revealing insights into non-affine responses and force fluctuations.

Contribution

It introduces a novel master equation approach that accounts for both contacts and gaps, advancing understanding of force network evolution in jammed particles.

Findings

Force fluctuations follow Gaussian statistics

Interparticle gaps exhibit multi-scale correlations

Mean force changes reflect non-affine responses

Abstract

Employing molecular dynamics simulations of jammed soft particles, we study microscopic responses of force-chain networks to quasi-static isotropic (de)compressions. We show that not only contacts but also interparticle gaps between the nearest neighbors must be considered for the stochastic evolution of the probability distribution functions (PDFs) of forces, where the mutual exchange of contacts and interparticle gaps, i.e. opening and closing contacts, are also crucial to the incremental system behaviors. By numerically determining the transition rates for all changes of contacts and gaps, we formulate a Master equation for the PDFs of forces, where the insight one gets from the transition rates is striking: The mean change of forces reflects non-affine system response, while their fluctuations obey uncorrelated Gaussian statistics. In contrast, interparticle gaps are reacting mostly affine in average, but imply multi-scale correlations according to a wider stable distribution function.