Time-reversal symmetries and equilibrium-like Langevin equations

Lokrshi Prawar Dadhichi, Klaus Kroy

Published 2022-10-25Version 1

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. Here we show that the resulting equilibrium form of the Langevin equation is associated with a nonequilibrium Hamiltonian and ask how precisely the broken equilibrium condition manifests itself therein. We find that this Hamiltonian is not time reversal invariant and that 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. Finally, this structure gives rise to a new, physically pertinent instance of frenesy.