Megías Alberto, Santos Andrés
Departamento de Física, Universidad de Extremadura, E-06006 Badajoz, Spain.
Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, E-06006 Badajoz, Spain.
Entropy (Basel). 2020 Nov 17;22(11):1308. doi: 10.3390/e22111308.
Finding the proper entropy-like Lyapunov functional associated with the inelastic Boltzmann equation for an isolated freely cooling granular gas is a still unsolved challenge. The original -theorem hypotheses do not fit here and the -functional presents some additional measure problems that are solved by the Kullback-Leibler divergence (KLD) of a reference velocity distribution function from the actual distribution. The right choice of the reference distribution in the KLD is crucial for the latter to qualify or not as a Lyapunov functional, the asymptotic "homogeneous cooling state" (HCS) distribution being a potential candidate. Due to the lack of a formal proof far from the quasielastic limit, the aim of this work is to support this conjecture aided by molecular dynamics simulations of inelastic hard disks and spheres in a wide range of values for the coefficient of restitution (α) and for different initial conditions. Our results reject the Maxwellian distribution as a possible reference, whereas they reinforce the HCS one. Moreover, the KLD is used to measure the amount of information lost on using the former rather than the latter, revealing a non-monotonic dependence with α.
找到与孤立自由冷却颗粒气体的非弹性玻尔兹曼方程相关的合适的类熵李雅普诺夫泛函仍然是一个未解决的挑战。原始的 - 定理假设在此不适用，并且 - 泛函存在一些额外的度量问题，这些问题通过参考速度分布函数相对于实际分布函数的库尔贝克 - 莱布勒散度（KLD）得以解决。KLD中参考分布的正确选择对于其是否能作为李雅普诺夫泛函至关重要，渐近“均匀冷却状态”（HCS）分布是一个潜在候选。由于远离准弹性极限缺乏形式证明，这项工作的目的是通过对恢复系数（α）的广泛取值以及不同初始条件下的非弹性硬磁盘和球体进行分子动力学模拟来支持这一猜想。我们的结果否定了麦克斯韦分布作为可能的参考，而强化了HCS分布。此外，KLD用于测量使用前者而非后者时丢失的信息量，揭示了其与α的非单调依赖性。
