Breaking microscopic reversibility with Lévy flights

¹ M. Smoluchowski Institute of Physics, Jagiellonian University - ul. Łojasiewicza 11, 30-348 Kraków, Poland
² Akhiezer Institute for Theoretical Physics, Kharkov Institute of Physics and Technology 61108 Kharkov, Ukraine
³ Max-Planck Institute for the Physics of Complex Systems - 01187 Dresden, Germany
⁴ Department of Physics and Astronomy “Galileo Galilei”, University of Padova - via Marzolo 8, 35122 Padova, Italy
⁵ Department of Physics, East Carolina University - Greenville, NC 27858, USA

Received: 16 May 2016
Accepted: 27 June 2016

Abstract

A system at equilibrium exhibits microscopic reversibility, i.e. any path in phase space is just as often traversed in one direction as that it is traversed in the opposite direction. We show how it is justified to characterize white Gaussian noise as equilibrium noise: when an overdamped particle in a potential is subjected to such noise, microscopic reversibility can be proven for most-probable-paths that lead from one potential well to another. However, when the overdamped particle is subjected to white Lévy noise, time-reversal symmetry is broken and microscopic reversibility is violated, even when the noise is symmetric. We, furthermore, derive how for an overdamped particle inside a parabolic potential microscopic reversibility is violated in the presence of Lévy white noise. Similar to Brownian vortexes, Lévy flights can be associated with the presence of Lévy vortexes in phase space.