Department of Biomedical Engineering, Oregon Health and Science University, Portland, Oregon 97239, USA.
Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523, USA.
J Chem Phys. 2019 Nov 7;151(17):174108. doi: 10.1063/1.5120511.
Probability currents are fundamental in characterizing the kinetics of nonequilibrium processes. Notably, the steady-state current J for a source-sink system can provide the exact mean-first-passage time (MFPT) for the transition from the source to sink. Because transient nonequilibrium behavior is quantified in some modern path sampling approaches, such as the "weighted ensemble" strategy, there is strong motivation to determine bounds on J-and hence on the MFPT-as the system evolves in time. Here, we show that J is bounded from above and below by the maximum and minimum, respectively, of the current as a function of the spatial coordinate at any time t for one-dimensional systems undergoing overdamped Langevin (i.e., Smoluchowski) dynamics and for higher-dimensional Smoluchowski systems satisfying certain assumptions when projected onto a single dimension. These bounds become tighter with time, making them of potential practical utility in a scheme for estimating J and the long time scale kinetics of complex systems. Conceptually, the bounds result from the fact that extrema of the transient currents relax toward the steady-state current.
概率流在描述非平衡过程的动力学中起着重要作用。值得注意的是，对于源汇系统，稳态电流 J 可以提供从源到汇的跃迁的精确平均首次通过时间 (MFPT)。由于瞬态非平衡行为在一些现代路径抽样方法中得到了量化，例如“加权集合”策略，因此强烈需要确定 J 的界限——因此也确定了 MFPT 的界限——随着系统随时间的演化。在这里，我们表明，对于一维系统经历过阻尼朗之万（即斯莫卢霍夫斯基）动力学和在投影到单个维度时满足某些假设的更高维斯莫卢霍夫斯基系统，在任何时间 t，J 分别被其作为空间坐标的函数的最大值和最小值所上下限。这些界限随着时间的推移而变得更加紧密，这使得它们在估计 J 和复杂系统的长时间尺度动力学的方案中具有潜在的实际用途。从概念上讲，这些界限是由于瞬态电流的极值朝着稳态电流松弛的事实而产生的。
