For a stochastic system, its evolution from one state to another can have a large number of possible paths. Non-uniformity in the field of system variables leads the local dynamics in state transition varies considerably from path to path and thus the distribution of the paths affects statistical characteristics of the system. Such a characteristic can be referred to as path-dependence of a system, and long-time correlation is an intrinsic feature of path-dependence systems. We employed a local path density operator to describe the distribution of state transition paths, and based on which we derived a new kinetic equation for path-dependent systems. The kinetic equation is similar in form to the Kramers-Moyal expansion, but with its expansion coefficients determined by the cumulants with respect to state transition paths, instead of transition moments. This characteristic makes it capable of accounting for the non-local feature of systems, which is essential in studies of large scale systems where the path-dependence is prominent. Short-time correlation approximation is also discussed. It shows that the cumulants of state transition paths are equivalent to jump moments when correlation time scales are infinitesimal, as makes the kinetic equation derived in this paper has the same physical consideration of the Fokker-Planck equation for Markov processes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.