Effective driven dynamics for onedimensional conditioned Langevin processes in the weaknoise limit
Abstract
In this work we focus on fluctuations of timeintegrated observables for a particle diffusing in a onedimensional periodic potential in the weaknoise asymptotics. Our interest goes to rare trajectories presenting an atypical value of the observable, that we study through a biased dynamics in a largedeviation framework. We determine explicitly the effective probabilityconserving dynamics which makes rare trajectories of the original dynamics become typical trajectories of the effective one. Our approach makes use of a weaknoise pathintegral description in which the action is minimised by the rare trajectories of interest. For ‘currenttype’ additive observables, we find criteria for 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, whose singular behaviour is between first and second order. 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 weaknoise and the largetime limits commute in this problem. Finally, we show how the biased dynamics can be mapped in practice to an explicit effective driven dynamics, which takes the form of a driven Langevin dynamics in an effective potential. The nontrivial 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.
 Publication:

Journal of Statistical Mechanics: Theory and Experiment
 Pub Date:
 January 2019
 DOI:
 10.1088/17425468/aaeda3
 arXiv:
 arXiv:1807.06438
 Bibcode:
 2019JSMTE..01.3201T
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 J. Stat. Mech. (2019) 013201