Non-adiabatic entropy production for non-Markov dynamics

Centro Atomico Bariloche and Instituto Balseiro, 8400 S. C. de Bariloche, ArgentinaWe extend the de nition of non-adiabatic entropy production given for Markovian systems in [M.Esposito and C. Van den Broeck, Phys. Rev. Lett. 104 090601, (2010)], to arbitrary non-Markovergodic dynamics. We also introduce a notion of stability characterizing non-Markovianity. Forstable non-Markovian systems, the non-adiabatic entropy production satis es an integral uctuationtheorem, leading to the second law of thermodynamics for transitions between non-equilibriumsteady-states. This quantity can also be written as a sum of products of generalized uxes andforces, thus being suitable for thermodynamics. On the other hand, the generalized uctuation-dissipation relation also holds, clarifying that the conditions for it to be satis ed are ergodicityand stability instead of Markovianity. We show that in spite of being counter-intuitive, the stabilitycriterion introduced in this work may be violated in non-Markovian systems even if they are ergodic,leading to a violation of the uctuation theorem and the generalized uctuation-dissipation relation.Stability represents then a necessary condition for the above properties to hold and explains whythe generalized uctuation-dissipation relation has remained elusive in the study of non-Markovsystems exhibiting non-equilibrium steady-states.