软相互作用中混沌的起源和非遍历性特征
Origin of chaos in soft interactions and signatures of nonergodicity
摘要：
讨论了一维盒子中两个粒子的硬点碰撞和软碰撞中混沌运动的出现。已知对于两个粒子的特定质量比$γ=m2/m1$，可以在硬点碰撞中获得遍历性，并且李雅普诺夫指数为零。然而，如果一个汤川粒子之间的相互作用被引入，我们分析表明，积极的李雅普诺夫指数产生的双重碰撞靠近墙壁。虽然最大的有限时间李雅普诺夫指数随着$γ$平滑地变化，但从初始条件下的有限时间李雅普诺夫指数的分布中提取的最可能的一个的出现次数揭示了相空间动力学的细节。特别是，可积和pseudointegrable动力学的影响，没有汤川相互作用的特定质量比可以清楚地确定，并证明了作为相空间探针的有限时间李雅普诺夫指数的灵敏度。不限于二维问题，如庞加莱截面，最可能的李雅普诺夫指数的出现次数表明自己作为一个合适的工具，以表征相空间动力学在更高的维度。这是显示的问题的两个相互作用的粒子在一个圆形台球。
英文摘要：
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 $γ=m2/m1$ 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 $γ$, 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.
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