自由冷却颗粒气体的Kullback-Leibler发散
Kullback--Leibler Divergence of a Freely Cooling Granular Gas
摘要：
对于孤立的自由冷却颗粒气体，找到与非弹性玻尔兹曼方程相关联的适当的类熵李雅普诺夫泛函仍然是一个未解决的挑战。原来的$H$定理假设不适合这里和$H$功能提出了一些额外的措施的问题，解决了从实际分布的参考速度分布函数的Kullback-Leibler分歧（KLD）。在KLD的参考分布的正确选择是至关重要的，后者有资格或不作为一个李雅普诺夫功能，渐近“均匀冷却状态”（HCS）分布是一个潜在的候选人。由于缺乏一个正式的证明远离准弹性极限，这项工作的目的是支持这一猜想的分子动力学模拟的非弹性硬盘和球体在很宽的范围内的恢复系数（$α$）和不同的初始条件。我们的结果拒绝麦克斯韦分布作为一个可能的参考，而他们加强了HCS之一。此外，KLD用于测量使用前者而不是后者时丢失的信息量，揭示了与$α$的非单调依赖性。
英文摘要：
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 $H$-theorem hypotheses do not fit here and the $H$-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 $α$.
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