On Effective Stochastic Generators for Conditioned Dynamics at an Atypical Reaction-Diffusion Current

TL;DR

This paper investigates how to modify stochastic particle systems to make rare current fluctuations typical, identifying conditions where the effective process retains the same local rules as the original.

Contribution

It derives conditions under which the effective stochastic process for atypical currents maintains the same local dynamics as the original process.

Findings

Effective process can have local dynamics under certain conditions.

Conditions identified for when the effective process matches the original process.

Approach generalizes to any time-integrated observable.

Abstract

We consider the fluctuations of a time-integrated particle current around an atypical value in a generic stochastic Markov process involving classical particles with two-site interaction and hardcore repulsion on a finite one-dimensional lattice with open boundaries. We address the question of which interactions one has to impose on such process to make the atypical value of the current typical. It is known that a corresponding effective stochastic Markov process might exist whose typical value of the current is equal to the atypical value of the current in the original process within a time-translational invariant regime. This effective process has, in principle, non-local transition rates. Nevertheless, it turns out that under some conditions the stochastic generator of the effective process has the same dynamical rules as the stochastic generator of the original process. We find these conditions and show that our approach can be generalized to any time-integrated observable.