# Structural Evolution of Granular Systems: Theory

## TL;DR

This paper develops a comprehensive theoretical framework for understanding how the structure of planar granular systems evolves over time, including steady states and stability, validated by simulations.

## Contribution

It introduces a general theory for cell order distribution evolution in granular systems, including dynamic equations and stability analysis, extending to structural organization.

## Key findings

- All steady states are stable.
- Steady states satisfy detailed balance for cell order ≤ 6.
- The theory is validated against extensive simulations.

## Abstract

A general theory is developed for the evolution of the cell order (CO) distribution in planar granular systems. Dynamic equations are constructed and solved in closed form for several examples: systems under compression; dilation of very dense systems; and the general approach to steady state. We find that all the steady states are stable and that they satisfy detailed balance-like condition when the CO$\,\leq 6$. Illustrative numerical solutions of the evolution are shown. Our theoretical results are validated against an extensive simulation of a sheared system. The formalism can be readily extended to other structural characteristics, paving the way to a general theory of structural organisation of granular systems.