Abstract
We revisit the Ornstein–Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic ‘equation of state’ of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.
Highlights
Gaussian fluctuation theory is one of the most successful branches of equilibrium statistical mechanics [1, 2]
In recent years, taking stochastic process rigorously developed by Kolmogorov as the mathematical representation, stochastic thermodynamics has emerged as the finitetime thermodynamic theory of mesoscopic systems, near and far from equilibrium [13, 14, 15, 16]
The fundamental aspects of this new development are the mathematical notion of stochastic entropy production [17, 18, 19], novel thermodynamic relationships collectively known as nonequilibrium work equalities, and fluctuation theorems [20, 21, 22, 23, 24], and the mathematical concept of non-equilibrium steady-state [25, 26, 27]
Summary
Gaussian fluctuation theory is one of the most successful branches of equilibrium statistical mechanics [1, 2]. The fundamental aspects of this new development are the mathematical notion of stochastic entropy production [17, 18, 19], novel thermodynamic relationships collectively known as nonequilibrium work equalities, and fluctuation theorems [20, 21, 22, 23, 24], and the mathematical concept of non-equilibrium steady-state [25, 26, 27]. Fundamental to all these advances is the notion of time reversal.
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