Non-adiabatic entropy production for non-Markov dynamics

TL;DR

This paper generalizes the concept of non-adiabatic entropy production to non-Markov ergodic systems, introduces a stability criterion, and explores its implications for fluctuation theorems and thermodynamics in non-Markovian dynamics.

Contribution

It extends the definition of non-adiabatic entropy production to non-Markovian systems and introduces a stability notion crucial for thermodynamic relations.

Findings

Non-adiabatic entropy production satisfies an integral fluctuation theorem in stable non-Markovian systems.

Stability is necessary for the fluctuation theorem and fluctuation-dissipation relation to hold.

Violations of stability can lead to violations of fundamental thermodynamic relations in non-Markovian systems.

Abstract

We extend the definition 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-Markov ergodic dynamics. We also introduce a notion of stability characterizing non-Markovianity. For stable non-Markovian systems, the non-adiabatic entropy production satisfies an integral fluctuation theorem, leading to the second law of thermodynamics for transitions between non-equilibrium steady-states. This quantity can also be written as a sum of products of generalized fluxes and forces, thus being suitable for thermodynamics. On the other hand, the generalized fluctuation-dissipation relation also holds, clarifying that the conditions for it to be satisfied are ergodicity and stability instead of Markovianity. We show that in spite of being counter-intuitive, the stability criterion introduced in this work may be violated in non-Markovian systems even if they are ergodic, leading to a violation of the fluctuation theorem and the generalized fluctuation-dissipation relation. Stability represents then a necessary condition for the above properties to hold and explains why the generalized fluctuation-dissipation relation has remained elusive in the study of non-Markov systems exhibiting non-equilibrium steady-states.