A Rigorous Foundation for Stochastic Thermodynamics via the Microcanonical Ensemble

Abstract

We consider a small Hamiltonian system strongly interacting with a much larger Hamiltonian system (the bath), while being driven by both a time-dependent control parameter and non-conservative forces. The joint system is assumed to be thermally isolated. Under the assumption of time-scale separation (TSS)--where the bath equilibrates much faster than the system and the external driving--the bath remains in instantaneous equilibrium, described by the microcanonical ensemble conditioned on the system state and the control parameter. We identify a decomposition of the total Hamiltonian that renders the bath energy an adiabatic invariant under slow evolution. This same decomposition defines the system Hamiltonian as the Hamiltonian of mean force, and ensures that neither the system nor the control parameter does reactive work on the bath. Using time-reversal symmetry and TSS, and without invoking any model details, we rigorously prove that the reduced dynamics of the system is Markovian and satisfies a form of local detailed balance (LDB) which involves transition probabilities but not path probabilities. By working entirely within the microcanonical framework and adopting a precise decomposition of the total energy, we provide rigorous definitions of bath entropy as the Boltzmann entropy, and of heat as the negative change of the bath energy. Our approach bypasses the ambiguities associated with conventional definitions of thermodynamic variables and path probabilities, and establishes a rigorous and thermodynamically consistent foundation for stochastic thermodynamics, valid even under strong system-bath coupling.