Stochastic Thermodynamics: Dissipativity, Losslessness, Accumulativity, Energy Storage, and Entropy Production

In this paper, we develop an energy-based dynamical system model driven by a Markov input process to present a unified framework for stochastic thermo-dynamics predicated on a stochastic dynamical systems formalism. Specifically, using a stochastic dissipativity, losslessness, and accumulativity theory, we develop a nonlinear stochastic port-Hamiltonian system model characterized by energy conservation and entropy nonconservation laws that are consistent with statistical thermodynamic principles. In particular, we show that the difference between the stored system energy and the supplied system energy for our stochastic thermodynamic model is a martingale with respect to the system filtration, whereas the system entropy is a submartingale with respect to the system filtration.