不连续布朗动力学的算法
Algorithms for Brownian dynamics across discontinuities
摘要：
分层系统中的质量扩散问题与不同科学学科中的应用有关，例如，化学、材料科学、土壤科学和生物医学工程。这些类型的模型系统中的数学挑战是匹配每一层中的时间相关扩散方程的解，使得满足它们之间的界面处的边界条件。随着层数的增加，解决方案可能变得越来越复杂。在这里，我们描述了一种替代的计算方法，多层扩散问题，这是基于描述的过阻尼布朗运动的粒子通过欠阻尼朗之万方程。在这种方法中，概率分布函数是从一个独立的单粒子轨迹的合奏的统计计算。为了允许在分层系统中的朗之万动力学的模拟，数值积分器必须补充的过渡算法跨越不连续的接口。三种常见类型的不连续性的算法：（一）摩擦系数的不连续性，（二）半透膜，和（三）阶梯函数化学势。一般情况下的接口，所有三个不连续存在（Kedem-Katchalsky边界）进行了讨论。我们证明了衍生算法的有效性和准确性，通过考虑一个简单的两层模型系统和比较朗之万动力学统计与解析解和替代计算结果。
英文摘要：
The problem of mass diffusion in layered systems has relevance to applications in different scientific disciplines, e.g., chemistry, material science, soil science, and biomedical engineering. The mathematical challenge in these type of model systems is to match the solutions of the time-dependent diffusion equation in each layer, such that the boundary conditions at the interfaces between them are satisfied. As the number of layers increases, the solutions may become increasingly complicated. Here, we describe an alternative computational approach to multi-layer diffusion problems, which is based on the description of the overdamped Brownian motion of particles via the underdamped Langevin equation. In this approach, the probability distribution function is computed from the statistics of an ensemble of independent single particle trajectories. To allow for simulations of Langevin dynamics in layered systems, the numerical integrator must be supplemented with algorithms for the transitions across the discontinuous interfaces. Algorithms for three common types of discontinuities are presented: (i) A discontinuity in the friction coefficient, (ii) a semi-permeable membrane, and (iii) a step-function chemical potential. The general case of an interface where all three discontinuities are present (Kedem-Katchalsky boundary) is also discussed. We demonstrate the validity and accuracy of the derived algorithms by considering a simple two-layer model system and comparing the Langevin dynamics statistics with analytical solutions and alternative computational results.
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