Origin of chaos in soft interactions and signatures of nonergodicity

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 Poincaré 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.

arXiv comment8 pages, 15 figures

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• Gauss map and Lyapunov exponents of interacting particles in a billiard
• Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems
• Classical Dynamics of Harmonically Trapped Interacting Particles
• Three unequal masses on a ring and soft triangular billiards
• Chaotic properties of Coulomb-interacting circular billiards
• Equivalence of kinetic-theory and random-matrix approaches to Lyapunov spectra of hard-sphere systems