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=m(2)/m(1) 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.

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands

“…We anticipate that this would be especially useful in its higher-dimensional generalizations [20]. A first study appears to support our expectations [26].…”

Characterizing the dynamics of higher dimensional nonintegrable conservative systems

“…Instead, we will analyze the following four measures: a.) The number of occurencies of the most probable FTLEs P(Λ p n , K) ≡ P Λ (K) which informs how many initial conditions (normalized) lead to the mode of the distribution [22,23,24]. P Λ (K) = 1 means that for all initial conditions the FTLEs are equal (within the precision 10 −3 ).…”

Intrinsic stickiness and chaos in open integrable billiards: Tiny border effects

“…Therefore in recent years more and more attention has been given to the description of particles confined inside boundaries (or billiards) which present some specific edges, softness etc. To mention some examples we have the edge roughness in quantum dots [1], unusual boundary conditions in two-dimensional billiards [2,3], effects of soft walls [4] and edge collisions [5] of interacting particles in a 1D billiard, rounding edge [6] and edge corrections [7] in a resonator, deformation of dielectric cavities [8], edge diffractions and the corresponding semiclassical quantization [9,10], among others.…”