Tanaka Shigenori
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan.
J Chem Phys. 2016 Mar 7;144(9):094103. doi: 10.1063/1.4942861.
A computational scheme to describe the temporal evolution of thermodynamic functions in stochastic nonequilibrium processes of isothermal classical systems is proposed on the basis of overdamped Langevin equation under given potential and temperature. In this scheme the associated Fokker-Planck-Smoluchowski equation for the probability density function is transformed into the imaginary-time Schrödinger equation with an effective Hamiltonian. The propagator for the time-dependent wave function is expressed in the framework of the path integral formalism, which can thus represent the dynamical behaviors of nonequilibrium molecular systems such as those conformational changes observed in protein folding and ligand docking. The present study then employs the diffusion Monte Carlo method to efficiently simulate the relaxation dynamics of wave function in terms of random walker distribution, which in the long-time limit reduces to the ground-state eigenfunction corresponding to the equilibrium Boltzmann distribution. Utilizing this classical-quantum correspondence, we can describe the relaxation processes of thermodynamic functions as an approach to the equilibrium state with the lowest free energy. Performing illustrative calculations for some prototypical model potentials, the temporal evolutions of enthalpy, entropy, and free energy of the classical systems are explicitly demonstrated. When the walkers initially start from a localized configuration in one- or two-dimensional harmonic or double well potential, the increase of entropy usually dominates the relaxation dynamics toward the equilibrium state. However, when they start from a broadened initial distribution or go into a steep valley of potential, the dynamics are driven by the decrease of enthalpy, thus causing the decrease of entropy associated with the spatial localization. In the cases of one- and two-dimensional asymmetric double well potentials with two minimal points and an energy barrier between them, we observe a nonequilibrium behavior that the system entropy first increases with the broadening of the initially localized walker distribution and then it begins to decrease along with the trapping at the global minimum of the potential, thus leading to the minimization of the free energy.
基于给定势和温度下的过阻尼朗之万方程，提出了一种用于描述等温经典系统随机非平衡过程中热力学函数时间演化的计算方案。在该方案中，概率密度函数的相关福克 - 普朗克 - 斯莫卢霍夫斯基方程被转化为具有有效哈密顿量的虚时薛定谔方程。含时波函数的传播子用路径积分形式表示，因此可以描述非平衡分子系统的动力学行为，如蛋白质折叠和配体对接中观察到的构象变化。本研究随后采用扩散蒙特卡罗方法，根据随机游走者分布有效地模拟波函数的弛豫动力学，在长时间极限下，该分布会简化为对应于平衡玻尔兹曼分布的基态本征函数。利用这种经典 - 量子对应关系，我们可以将热力学函数的弛豫过程描述为向具有最低自由能的平衡态的趋近。通过对一些典型模型势进行说明性计算，明确展示了经典系统焓、熵和自由能的时间演化。当游走者最初从一维或二维谐波或双阱势中的局域构型开始时，熵的增加通常主导着向平衡态的弛豫动力学。然而，当它们从拓宽的初始分布开始或进入势的陡峭谷底时，动力学由焓的降低驱动，从而导致与空间局域化相关的熵的降低。在具有两个极小点且它们之间有能垒的一维和二维不对称双阱势的情况下，我们观察到一种非平衡行为，即系统熵首先随着最初局域化的游走者分布的拓宽而增加，然后随着在势的全局最小值处的捕获而开始降低，从而导致自由能最小化。
