Instituto de Física, Universidade de São Paulo, São Paulo CEP 05508-090, SP, Brazil.
Instituto de Astronomia Geofísica e Ciências Atmosféricas, Universidade de São Paulo, São Paulo CEP 05508-090, SP, Brazil.
Chaos. 2019 Oct;29(10):103122. doi: 10.1063/1.5120023.
Numerical experiments of the statistical evolution of an ensemble of noninteracting particles in a time-dependent billiard with inelastic collisions reveals the existence of three statistical regimes for the evolution of the speed ensemble, namely, diffusion plateau, normal growth/exponential decay, and stagnation. These regimes are linked numerically to the transition from Gauss-like to Boltzmann-like speed distributions. Furthermore, the different evolution regimes are obtained analytically through velocity-space diffusion analysis. From these calculations, the asymptotic root mean square of speed, initial plateau, and the growth/decay rates for an intermediate number of collisions are determined in terms of the system parameters. The analytical calculations match the numerical experiments and point to a dynamical mechanism for "thermalization," where inelastic collisions and a high-dimensional phase space lead to a bounded diffusion in the velocity space toward a stationary distribution function with a kind of "reservoir temperature" determined by the boundary oscillation amplitude and the restitution coefficient.
数值实验表明，在具有非弹性碰撞的时变双曲型系统中，相互作用粒子系综的统计演化存在三个速度系综演化的统计区，即扩散平台、正常增长/指数衰减和停滞。这些区与从高斯型到玻尔兹曼型速度分布的转变在数值上是相关的。此外，通过速度空间扩散分析，可以从分析上得到不同的演化区。通过这些计算，可以根据系统参数确定速度的渐近均方根、初始平台以及中间碰撞次数的增长/衰减率。分析计算与数值实验相匹配，并指出了一种“热化”的动力学机制，其中非弹性碰撞和高维相空间导致速度空间中的有界扩散，形成一个具有由边界振动幅度和恢复系数决定的“储层温度”的稳定分布函数。
