Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture

Probability currents are fundamental in characterizing the kinetics of non-equilibrium processes. Notably, the steady-state current $J_{ss}$ for a source-sink system can provide the exact mean-first-passage time (MFPT) for the transition from source to sink. Because transient non-equilibrium behavior is quantified in some modern path sampling approaches, such as the "weighted ensemble" strategy, there is strong motivation to determine bounds on $J_{ss}$ -- and hence on the MFPT -- as the system evolves in time. Here we show that $J_{ss}$ 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 over-damped 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 $J_{ss}$ and the long-timescale kinetics of complex systems. Conceptually, the bounds result from the fact that extrema of the transient currents relax toward the steady-state current.