Time-reversal symmetries and equilibrium-like Langevin equations

TL;DR

This paper explores how certain Langevin equations can resemble equilibrium systems through time-reversal symmetry, revealing insights into entropy production, Hamiltonian properties, and the role of noise in dissipation.

Contribution

It provides explicit analysis of how nonequilibrium Hamiltonians can lose time-reversal invariance and how reactive and dissipative fluxes behave under these conditions.

Findings

Reactive fluxes contribute to entropy production.

Time-reversal symmetry can be broken in the Hamiltonian structure.

Noise can be solely responsible for dissipation.

Abstract

Graham has shown in Z. Physik B 26, 397-405 (1977) that a fluctuation-dissipation relation can be imposed on a class of non-equilibrium Markovian Langevin equations that admit a stationary solution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin equation is associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight into how this Hamiltonian may loose its time reversal invariance and how the "reactive" and "dissipative" fluxes loose their distinct time reversal symmetries. The antisymmetric coupling matrix between forces and fluxes no longer originates from Poisson brackets and the "reactive" fluxes contribute to the ("housekeeping") entropy production, in the steady state. The time-reversal even and odd parts of the nonequilibrium Hamiltonian contribute in qualitatively different but physically instructive ways to the entropy. We find instances where fluctuations due to noise are solely responsible for the dissipation. Finally, this structure gives rise to a new, physically pertinent instance of frenesy.