# Contact forces distribution for a granular material from a Monte Carlo study on a single grain

## TL;DR

This study introduces a Monte Carlo method to analyze force distributions on a single grain in granular materials, revealing gamma-distributed pressures and an equipartition relation, advancing understanding of force networks at the grain level.

## Contribution

Develops a Monte Carlo sampling technique for the force ensemble on a single grain, demonstrating gamma-distributed pressures and an equipartition theorem in granular systems.

## Key findings

- Pressure per grain follows a gamma distribution.
- Average pressure satisfies an equipartition relation.
- Force variables are independent and exponentially distributed.

## Abstract

The force network ensemble is one of the most promising statistical descriptions of granular media, with an entropy accounting for all force configurations at mechanical equilibrium consistent with some external stress. It is possible to define a temperature-like parameter, the angoricity {\alpha}^{-1}, which under isotropic compression is a scalar variable. This ensemble is frequently studied on whole packings of grains; however, previous works have shown that spatial correlations can be neglected in many cases, opening the door to studies on a single grain. Our work develops a Monte Carlo method to sample the force ensemble on a single grain at constant angoricity on two and three-dimensional mono-disperse granular systems, both with or without static friction. The results show that, despite the steric exclusions and the constrictions of Coulomb's limit and repulsive normal forces, the pressure per grain always show a gamma distribution with scale parameter {\nu} = {\alpha}^{-1} and shape parameter k close to k', the number of degrees of freedom in the system. Moreover, the average pressure per grain fulfills an equipartition theorem <p>=k'{\alpha}^{-1} in all cases (in close parallelism with the one for an ideal gas). These results suggest the existence of k' independent random variables (i.e. elementary forces) with identical exponential distributions as the basic elements for describing the force network ensemble at low angoricities under isotropic compression, in analogy with the volume ensemble of granular materials.