Thermal equilibrium with generalized time-reversal symmetry

D.Lucente (Department of Mathematics & Physics, University of Campania “Luigi Vanvitelli”)

Determining whether a stochastic system is at equilibrium requires identifying the correct transformation properties of dynamical variables under time reversal. This is essential to assess the validity of detailed balance and to define the entropy production rate unambiguously.

I will present a framework to construct equilibrium Langevin dynamics from their reversible deterministic counterparts. Building on this, I will show that for systems with multiple timescales, slow degrees of freedom evolve, in a suitable limit, according to a Langevin equation satisfying the generalized form of detailed balance. I will then discuss a criterion to assess whether a given stochastic dynamics is at thermal equilibrium, applicable to all 2-dimensional systems and, more generally, to systems admitting an integrable deterministic limit. In these cases, the time-reversal symmetry is naturally identified in action-angle variables.

As an illustration, I will discuss the Lotka-Volterra model, where a stochastic process with a non-trivial parity rule can be explicitly constructed.