From pinned billiard balls to partial differential equations

TL;DR

This paper models energy transfer in a large system of fixed billiard balls using stochastic mean-field assumptions, deriving nonlinear partial difference equations and validating results through numerical simulations.

Contribution

It introduces a novel stochastic mean-field framework for analyzing energy propagation in fixed billiard systems, leading to new coupled nonlinear partial difference equations.

Findings

Derivation of coupled nonlinear partial difference equations for energy propagation.

Numerical simulations demonstrating the validity of the theoretical model.

Insights into energy transfer mechanisms in fixed billiard systems.

Abstract

We discuss the propagation of kinetic energy through billiard balls fixed in place along a one-dimensional segment. The number of billiard balls is assumed to be large but finite and we assume kinetic energy propagates following the usual collision laws of physics. Assuming an underlying stochastic mean-field for the expectation and the variance of the kinetic energy, we derive a coupled system of nonlinear partial difference equations. Our results are illustrated by numerical simulations.