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arXiv 1804.01285cond-mat.stat-mech
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含时Fokker-Planck方程的物理相容数值解

Physically consistent numerical solver for time-dependent Fokker-Planck equations

Viktor Holubec, Klaus Kroy, Stefano Steffenoni

备注：
20 pages, 8 figures
发表信息：
Phys. Rev. E 99, 032117 (2019)
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摘要：

本文提出了一种求解过阻尼随机过程的含时Fokker-Planck方程（FPE）的简单的物理相容方法，也称为Smoluchowski方程。它产生过渡和稳态行为，并允许计算的时刻生成和大偏差函数的观测定义沿随机轨迹，如波动的粒子电流，热量和工作。其关键策略是通过一个主方程来近似FPE，该主方程在位形空间中的转移率服从局部细致平衡条件，以实现任意离散化。它的时间相关的解决方案是通过直接计算的时间顺序的指数，代表的FPE的传播，通过在离散空间中的所有可能的路径求和。因此，该方法不仅保留了正性和归一化的解决方案，但也产生了物理上合理的总熵产生，无论离散。为了证明该方法的有效性，并验证其应用潜力，我们将其与布朗动力学模拟的热机的基础上，一个活跃的布朗粒子被困在一个随时间变化的四次势。

英文摘要：

We present a simple thermodynamically consistent method for solving time-dependent Fokker--Planck equations (FPE) for over-damped stochastic processes, also known as Smoluchowski equations. It yields both transition and steady-state behavior and allows for computations of moment-generating and large-deviation functions of observables defined along stochastic trajectories, such as the fluctuating particle current, heat and work. The key strategy is to approximate the FPE by a Master equation with transition rates in configuration space that obey a local detailed balance condition for arbitrary discretization. Its time-dependent solution is obtained by a direct computation of the time-ordered exponential, representing the propagator of the FPE, by summing over all possible paths in the discretized space. The method thus not only preserves positivity and normalization of the solutions, but also yields a physically reasonable total entropy production, regardless of the discretization. To demonstrate the validity of the method and to exemplify its potential for applications, we compare it against Brownian-dynamics simulations of a heat engine based on an active Brownian particle trapped in a time-dependent quartic potential.

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