Soft wall effects on interacting particles in billiards

TL;DR

This study investigates how soft, physically realizable wall potentials influence the dynamics of two interacting particles in a 1D billiard, revealing significant effects on chaos, regular islands, and ergodicity as wall softness varies.

Contribution

It introduces a continuous model of wall softness in 1D billiards, linking soft walls to a soft triangular billiard and analyzing their impact on particle dynamics and chaos.

Findings

Small wall softness reduces chaos and increases regular islands.

Lyapunov exponent decreases with increasing wall softness.

Phase-space dynamics become more ergodic with softer walls.

Abstract

The effect of physically realizable wall potentials (soft walls) on the dynamics of two interacting particles in a one-dimensional (1D) billiard is examined numerically. The 1D walls are modeled by the error function and the transition from hard to soft walls can be analyzed continuously by varying the softness parameter $\sigma$. For $\sigma\to 0$ the 1D hard wall limit is obtained and the corresponding wall force on the particles is the $\delta$-function. In this limit the interacting particle dynamics agrees with previous results obtained for the 1D hard walls. We show that the two interacting particles in the 1D soft walls model is equivalent to one particle inside a soft right triangular billiard. Very small values of $\sigma$ substantiously change the dynamics inside the billiard and the mean finite-time Lyapunov exponent decreases significantly as the consequence of regular islands which appear due to the low-energy double collisions (simultaneous particle-particle-1D wall collisions). The rise of regular islands and sticky trajectories induced by the 1D wall softness is quantified by the number of occurrences of the most probable finite-time Lyapunov exponent. On the other hand, chaotic motion in the system appears due to the high-energy double collisions. In general we observe that the mean finite-time Lyapunov exponent decreases when $\sigma$ increases, but the number of occurrences of the most probable finite-time Lyapunov exponent increases, meaning that the phase-space dynamics tends to be more ergodiclike. Our results suggest that the transport efficiency of interacting particles and heat conduction in periodic structures modeled by billiards will strongly be affected by the smoothness of physically realizable walls.