DNRF centre Glass and Time, IMFUFA, Department of Sciences, Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark.
J Chem Phys. 2012 Jun 14;136(22):224106. doi: 10.1063/1.4726728.
Classical Newtonian dynamics is analytic and the energy of an isolated system is conserved. The energy of such a system, obtained by the discrete "Verlet" algorithm commonly used in molecular dynamics simulations, fluctuates but is conserved in the mean. This is explained by the existence of a "shadow Hamiltonian" H [S. Toxvaerd, Phys. Rev. E 50, 2271 (1994)], i.e., a Hamiltonian close to the original H with the property that the discrete positions of the Verlet algorithm for H lie on the analytic trajectories of H. The shadow Hamiltonian can be obtained from H by an asymptotic expansion in the time step length. Here we use the first non-trivial term in this expansion to obtain an improved estimate of the discrete values of the energy. The investigation is performed for a representative system with Lennard-Jones pair interactions. The simulations show that inclusion of this term reduces the standard deviation of the energy fluctuations by a factor of 100 for typical values of the time step length. Simulations further show that the energy is conserved for at least one hundred million time steps provided the potential and its first four derivatives are continuous at the cutoff. Finally, we show analytically as well as numerically that energy conservation is not sensitive to round-off errors.
经典牛顿力学是解析的，孤立系统的能量是守恒的。在分子动力学模拟中常用的离散“Verlet”算法中，这样一个系统的能量会发生波动，但平均值是守恒的。这可以通过存在一个“影子哈密顿量”H[S. Toxvaerd, Phys. Rev. E 50, 2271 (1994)]来解释，即一个与原始 H 接近的哈密顿量，具有这样的性质，即 H 的 Verlet 算法的离散位置位于 H 的解析轨迹上。影子哈密顿量可以通过时间步长的渐近展开式从 H 中得到。在这里，我们使用这个展开式的第一个非平凡项来获得能量的离散值的改进估计。我们对一个具有 Lennard-Jones 对相互作用的代表性系统进行了研究。模拟表明，对于典型的时间步长值，包含这个项可以将能量波动的标准偏差降低 100 倍。模拟还表明，只要在截止处势能及其前四个导数是连续的，那么能量就可以至少保持 1 亿个时间步长的守恒。最后，我们通过分析和数值方法证明了能量守恒对舍入误差不敏感。
