Uline Mark J, Siderius Daniel W, Corti David S
School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-2100, USA.
J Chem Phys. 2008 Mar 28;128(12):124301. doi: 10.1063/1.2889939.
We consider various ensemble averages within the molecular dynamics (MD) ensemble, corresponding to those states sampled during a MD simulation in which the application of periodic boundary conditions imposes a constraint on the momentum of the center of mass. As noted by Shirts et al. [J. Chem. Phys. 125, 164102 (2006)] for an isolated system, we find that the principle of equipartition is not satisfied within such simulations, i.e., the total kinetic energy of the system is not shared equally among all the translational degrees of freedom. Nevertheless, we derive two different versions of Tolman's generalized equipartition theorem, one appropriate for the canonical ensemble and the other relevant to the microcanonical ensemble. In both cases, the breakdown of the principle of equipartition immediately follows from Tolman's result. The translational degrees of freedom are, however, still equivalent, being coupled to the same bulk property in an identical manner. We also show that the temperature of an isolated system is not directly proportional to the average of the total kinetic energy (in contrast to the direct proportionality that arises between the temperature of the external bath and the kinetic energy within the canonical ensemble). Consequently, the system temperature does not appear within Tolman's generalized equipartition theorem for the microcanonical ensemble (unlike the immediate appearance of the temperature of the external bath within the canonical ensemble). Both of these results serve to highlight the flaws in the argument put forth by Hertz [Ann. Phys. 33, 225 (1910); 33, 537 (1910)] for defining the entropy of an isolated system via the integral of the phase space volume. Only the Boltzmann-Planck entropy definition, which connects entropy to the integral of the phase space density, leads to the correct description of the properties of a finite, isolated system. We demonstrate that the use of the integral of the phase space volume leads to unphysical results, indicating that the property of adiabatic invariance has little to do with the behavior of small systems.
我们考虑分子动力学(MD)系综中的各种系综平均值,这些平均值对应于MD模拟过程中采样的那些状态,在该模拟中,周期性边界条件的应用对质心的动量施加了约束。正如Shirts等人[《化学物理杂志》125, 164102 (2006)]针对孤立系统所指出的,我们发现在此类模拟中能量均分原理并不成立,即系统的总动能并非在所有平动自由度之间平均分配。尽管如此,我们推导出了托尔曼广义能量均分定理的两个不同版本,一个适用于正则系综,另一个适用于微正则系综。在这两种情况下,能量均分原理的失效都直接源于托尔曼的结果。然而,平动自由度仍然是等效的,它们以相同的方式与相同的体性质相关联。我们还表明,孤立系统的温度与总动能的平均值并非直接成正比(这与正则系综中外加浴的温度与动能之间出现的直接比例关系形成对比)。因此,在微正则系综的托尔曼广义能量均分定理中不会出现系统温度(这与正则系综中外加浴温度的直接出现情况不同)。这两个结果都凸显了赫兹[《物理学年鉴》33, 225 (1910); 33, 537 (1910)]通过相空间体积积分来定义孤立系统熵的论证中的缺陷。只有将熵与相空间密度积分联系起来的玻尔兹曼 - 普朗克熵定义,才能对有限孤立系统的性质给出正确描述。我们证明,使用相空间体积积分会导致非物理结果,这表明绝热不变性的性质与小系统的行为几乎无关。