Different time scales in dynamic systems with multiple exits

TL;DR

This paper demonstrates that dynamic systems with multiple exits exhibit different time scales, such as mean exit times and fluxes, which are influenced by the presence of other exits, challenging single-exit assumptions.

Contribution

The study provides theoretical evidence and analytical methods to understand multiple time scales in systems with several exits, extending beyond single-exit models.

Findings

Existence of multiple time scales like mean exit times and fluxes.

Presence of other exits influences statistical properties.

Analytical and simulation methods confirm the theoretical predictions.

Abstract

Stochastic biochemical and transport processes have various final outcomes, and they can be viewed as dynamic systems with multiple exits. Many current theoretical studies, however, typically consider only a single time scale for each specific outcome, effectively corresponding to a single-exit process and assuming the independence of each exit process. But the presence of other exits influences the statistical properties and dynamics measured at any specific exit. Here, we present theoretical arguments to explicitly show the existence of different time scales, such as mean exit times and inverse exit fluxes, for dynamic processes with multiple exits. This implies that the statistics of any specific exit dynamics cannot be considered without taking into account the presence of other exits. Several illustrative examples are described in detail using analytical calculations, mean-field estimates, and kinetic Monte Carlo computer simulations. The underlying microscopic mechanisms for the existence of different time scales are discussed. The results are relevant for understanding the mechanisms of various biological, chemical, and industrial processes, including transport through channels and pores.