Effective driven dynamics for one-dimensional conditioned Langevin processes in the weak-noise limit

TL;DR

This paper investigates rare fluctuations in a one-dimensional particle system under weak noise, revealing a dynamical phase transition and providing a method to derive effective driven dynamics that explain these rare events.

Contribution

It introduces a new approach to explicitly determine effective driven Langevin dynamics for rare events in weak-noise systems, linking dynamical phase transitions to depinning phenomena.

Findings

Identification of propagative trajectories for large deviations

Development of a method to compute the scaled cumulant generating function

Mapping of biased dynamics to an effective driven Langevin process

Abstract

In this work we focus on fluctuations of time-integrated observables for a particle diffusing in a one-dimensional periodic potential in the weak-noise asymptotics. Our interest goes to rare trajectories presenting an atypical value of the observable, that we study through a biased dynamics in a large-deviation framework. We determine explicitly the effective probability-conserving dynamics which makes rare trajectories of the original dynamics become typical trajectories of the effective one. Our approach makes use of a weak-noise path-integral description in which the action is minimised by the rare trajectories of interest. For `current-type' additive observables, we find the emergence of a propagative trajectory minimising the action for large enough deviations, revealing the existence of a dynamical phase transition at a fluctuating level. In addition, we provide a new method to determine the scaled cumulant generating function of the observable without having to optimise the action. It allows one to show that the weak-noise and the large-time limits commute in this problem. Finally, we show how the biased dynamics can be mapped in practice to an effective driven dynamics, which takes the form of a driven Langevin dynamics in an effective potential. The non-trivial shape of this effective potential is key to understand the link between the dynamical phase transition in the large deviations of current and the standard depinning transition of a particle in a tilted potential.