Non-equilibrium statistical physics of systems with finite heat baths

In this presentation I analyze the thermodynamics of a system connected to a single, finite heat bath. I concentrate on the case where the work reservoir can only change the Hamiltonian of the system, not the Hamiltonian of the bath, nor the interaction Hamiltonian coupling the system to the bath. In order to apply the tools of conventional stochasteic thermodynamics, I assume that the system is coupled to a second, infinite heat reservoir, in addition to the finite heat bath. I prove that in this scenario the entropy production rate of the joint system-(finite)-bath is lower-bounded by the rate of change of the mutual information between the system and the finite bath. This means that while it is possible to use a semi-static process to simultaneously extract the no-nequilibrium free energy of the system considered by itself, together with the non-equilibrium free energy of the finite bath considered by itself, it is impossible to at the same time extract the free energy stored in the mutual information of those two systems.