TY - JOUR

T1 - Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture

AU - Copperman, Jeremy

AU - Aristoff, David

AU - Makarov, Dmitrii E.

AU - Simpson, Gideon

AU - Zuckerman, Daniel M.

N1 - Funding Information:
We are very appreciative of discussions with Sasha Berezhkovskii and David Zuckerman. This work was supported by the NSF, Grant Nos. CHE-1566001 (to D.E.M.), DMS-1522398 (D.A.), and DMS-1818726 (D.A. and G.S.), by the NIH, Grant No. GM115805 (D.M.Z.), and by the Robert A. Welch Foundation, Grant No. F-1514 (D.E.M.).
Publisher Copyright:
© 2019 Author(s).

PY - 2019/11/7

Y1 - 2019/11/7

N2 - Probability currents are fundamental in characterizing the kinetics of nonequilibrium processes. Notably, the steady-state current Jss 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 Jss - and hence on the MFPT - as the system evolves in time. Here, we show that Jss 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 Jss 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.

AB - Probability currents are fundamental in characterizing the kinetics of nonequilibrium processes. Notably, the steady-state current Jss 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 Jss - and hence on the MFPT - as the system evolves in time. Here, we show that Jss 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 Jss 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.

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U2 - 10.1063/1.5120511

DO - 10.1063/1.5120511

M3 - Article

C2 - 31703496

AN - SCOPUS:85074696988

VL - 151

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 17

M1 - 5120511

ER -