Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom.
Phys Rev Lett. 2014 Mar 7;112(9):098002. doi: 10.1103/PhysRevLett.112.098002. Epub 2014 Mar 5.
We present numerical simulations that allow us to compute the number of ways in which N particles can pack into a given volume V. Our technique modifies the method of Xu, Frenkel, and Liu [Phys. Rev. Lett. 106, 245502 (2011)] and outperforms existing direct enumeration methods by more than 200 orders of magnitude. We use our approach to study the system size dependence of the number of distinct packings of a system of up to 128 polydisperse soft disks. We show that, even though granular particles are distinguishable, we have to include a factor 1=N! to ensure that the entropy does not change when exchanging particles between systems in the same macroscopic state. Our simulations provide strong evidence that the packing entropy, when properly defined, is extensive. As different packings are created with unequal probabilities, it is natural to express the packing entropy as S = − Σ(i)p(i) ln pi − lnN!, where pi denotes the probability to generate the ith packing. We can compute this quantity reliably and it is also extensive. The granular entropy thus (re)defined, while distinct from the one proposed by Edwards [J. Phys. Condens. Matter 2, SA63 (1990)], does have all the properties Edwards assumed.
我们提出了数值模拟，可以计算 N 个粒子在给定体积 V 中填充的方式数量。我们的技术修改了 Xu、Frenkel 和 Liu 的方法[Phys. Rev. Lett. 106, 245502 (2011)]，比现有的直接枚举方法快 200 多个数量级。我们使用这种方法研究了多达 128 个多分散软磁盘系统的系统尺寸对不同填充方式数量的依赖性。我们表明，尽管颗粒是可区分的，但我们必须包含一个 1=N!的因子，以确保当在相同宏观状态下的系统之间交换粒子时，熵不会发生变化。我们的模拟提供了强有力的证据，表明在适当定义的情况下，填充熵是广泛的。由于不同的填充方式具有不同的概率，因此将填充熵表示为 S = -Σ(i)p(i)ln pi - lnN!是很自然的，其中 pi 表示生成第 i 种填充方式的概率。我们可以可靠地计算这个量，它也是广泛的。因此，重新定义的颗粒熵虽然与 Edwards [J. Phys. Condens. Matter 2, SA63 (1990)]提出的不同，但具有 Edwards 假设的所有特性。
