Origin of chaos in soft interactions and signatures of nonergodicity

TL;DR

This paper investigates how soft interactions induce chaos in a two-particle system in a one-dimensional box, revealing how Lyapunov exponents and their distributions can diagnose phase-space dynamics and nonergodicity.

Contribution

It analytically demonstrates that Yukawa interactions generate positive Lyapunov exponents and introduces the distribution of finite-time Lyapunov exponents as a phase-space diagnostic tool.

Findings

Yukawa interactions induce chaos near walls in soft collision systems.

Distribution of Lyapunov exponents reveals phase-space structures.

Finite-time Lyapunov exponent analysis extends to higher-dimensional systems.

Abstract

The emergence of chaotic motion is discussed for hard-point like and soft collisions between two particles in a one-dimensional box. It is known that ergodicity may be obtained in hard-point like collisions for specific mass ratios $\gamma=m2/m1$ of the two particles and that Lyapunov exponents are zero. However, if a Yukawa interaction between the particles is introduced, we show analytically that positive Lyapunov exponents are generated due to double collisions close to the walls. While the largest finite-time Lyapunov exponent changes smoothly with $\gamma$, the number of occurrences of the most probable one, extracted from the distribution of finite-time Lyapunov exponents over initial conditions, reveals details about the phase-space dynamics. In particular, the influence of the integrable and pseudointegrable dynamics without Yukawa interaction for specific mass ratios can be clearly identified and demonstrates the sensitivity of the finite-time Lyapunov exponents as a phase-space probe. Being not restricted to two-dimensional problems such as Poincar\'e sections, the number of occurrences of the most probable Lyapunov exponents suggests itself as a suitable tool to characterize phase-space dynamics in higher dimensions. This is shown for the problem of two interacting particles in a circular billiard.