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Stochastic Thermodynamics in a Non-Markovian Dynamical System

The developing field of stochastic thermodynamics extends concepts of macroscopic thermodynamics such as entropy production and work to the microscopic level of individual trajectories taken by a system through phase space. The scheme involves coupling the system to an environment - typically a source of Markovian noise that affects the dynamics of the system. Here we extend this framework to consider a non-Markovian environment, one whose dynamics have memory and which create additional correlations with the system variables, and illustrate this with a selection of simple examples. Such an environment produces a rich variety of behaviours. In particular, for a case of thermal relaxation, the distributions of entropy produced under the non-Markovian dynamics differ from the equivalent case of Markovian dynamics only by a delay time. When a time-dependent external work protocol is turned on, the system's correlations with the environment can either assist or hinder its approach to equilibrium, and affect its production of entropy, depending on the coupling strength between the system and environment.