Abstract
In this paper, we develop an energy-based, large-scale dynamical system model driven by Markov diffusion processes to present a unified framework for statistical thermodynamics predicated on a stochastic dynamical systems formalism. Specifically, using a stochastic state space formulation, we develop a nonlinear stochastic compartmental dynamical system model characterized by energy conservation laws that is consistent with statistical thermodynamic principles. In particular, we show that the difference between the average supplied system energy and the average stored system energy for our stochastic thermodynamic model is a martingale with respect to the system filtration. In addition, we show that the average stored system energy is equal to the mean energy that can be extracted from the system and the mean energy that can be delivered to the system in order to transfer it from a zero energy level to an arbitrary nonempty subset in the state space over a finite stopping time.
Highlights
In an attempt to generalize classical thermodynamics to irreversible nonequilibrium thermodynamics, a relatively new framework has been developed that combines stochasticity and nonequilibrium dynamics
Stochastic thermodynamics is applicable to nonequilibrium systems extending the validity of the laws of thermodynamics beyond the linear response regime by providing a system thermodynamic paradigm formulated on the level of individual system state realizations that are arbitrarily far from equilibrium
We combined thermodynamics and stochastic dynamical system theory to provide a system-theoretic foundation of thermodynamics
Summary
In an attempt to generalize classical thermodynamics to irreversible nonequilibrium thermodynamics, a relatively new framework has been developed that combines stochasticity and nonequilibrium dynamics. Einstein was the first to formulate the theory of Brownian motion by assuming that the particles suspended in the liquid contribute to the thermal fluctuations of the medium and, in accordance with the principle of equipartition of energy [14], the average translational kinetic energy of each particle [10]. The frictional forces extract the particle kinetic energy, which in turn is injected back to the particles in the form of thermal fluctuations This captures the phenomenological behavior of a Brownian particle suspended in a fluid medium which can be modeled as a continuous Markov process [17]. The entropy production of individual sample path trajectories of a stochastic thermodynamic system described by a Markov process is not restricted by the second law, but rather the average entropy production is determined to be positive. We show that the steady-state distribution of the large-scale sample path system energies is uniform, leading to system energy equipartitioning corresponding to a maximum entropy equilibrium state
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