扩散中的布朗模拟和单向通量
Brownian Simulations and Uni-Directional Flux in Diffusion
摘要：
布朗动力学模拟需要将小的离散模拟体积连接到保持在固定浓度和电压下的大浴。连续体浴通过接口连接到模拟，位于浴中足够远离通道。平均边界浓度必须通过在界面处注入和去除颗粒来维持在浴中的值。注入模拟体积的粒子表示单向扩散通量，而传出粒子表示相反方向的单向通量。经典的扩散方程定义了净扩散通量，而不是单向通量。根据维纳过程的经典扩散的随机公式导致维纳路径积分，它可以将净通量分解为单向通量。这些单向通量是无限的，尽管净通量是有限的，并且与经典理论一致。我们发现，无限的单向通量是一个工件所造成的替代朗之万动力学与Smoluchowski近似，这是经典的扩散。Smoluchowski近似在比朗之万方程的弛豫时间$1/γ$短的时间尺度上失败。我们发现的单向通量（源强度）需要保持平均边界浓度的方式与布朗粒子的物理。该单向通量与浓度成正比，与$\sqrt{Δt}$成反比。我们开发了一个BD模拟，保持固定的平均边界浓度的方式符合实际的物理界面，而不会产生虚假的边界层。
英文摘要：
Brownian dynamics simulations require the connection of a small discrete simulation volume to large baths that are maintained at fixed concentrations and voltages. The continuum baths are connected to the simulation through interfaces, located in the baths sufficiently far from the channel. Average boundary concentrations have to be maintained at their values in the baths by injecting and removing particles at the interfaces. The particles injected into the simulation volume represent a unidirectional diffusion flux, while the outgoing particles represent the unidirectional flux in the opposite direction. The classical diffusion equation defines net diffusion flux, but not unidirectional fluxes. The stochastic formulation of classical diffusion in terms of the Wiener process leads to a Wiener path integral, which can split the net flux into unidirectional fluxes. These unidirectional fluxes are infinite, though the net flux is finite and agrees with classical theory. We find that the infinite unidirectional flux is an artifact caused by replacing the Langevin dynamics with its Smoluchowski approximation, which is classical diffusion. The Smoluchowski approximation fails on time scales shorter than the relaxation time $1/γ$ of the Langevin equation. We find the unidirectional flux (source strength) needed to maintain average boundary concentrations in a manner consistent with the physics of Brownian particles. This unidirectional flux is proportional to the concentration and inversely proportional to $\sqrt{Δt}$ to leading order. We develop a BD simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics of the interface and without creating spurious boundary layers.
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