A nonequilibrium distribution for stochastic thermodynamics

TL;DR

This work introduces an extended Gibbs canonical distribution by incorporating an internal nonequilibrium variable $\xi(t)$, enabling a microscopic description of work and uncompensated heat within stochastic thermodynamics. By allowing energy levels $E_i(\lambda,\xi)$ to depend on both a control parameter $\lambda$ and the internal state $\xi$, and defining the stochastic work $\delta W_i=(\partial E_i/\partial \lambda)_{\xi} d\lambda$ and the stochastic uncompensated heat $\delta Q'_i=A_i d\xi$ with $A_i=- (\partial E_i/\partial \xi)_{\lambda}$, the framework recovers macroscopic laws upon averaging and yields new nonequilibrium identities. It derives a Jarzynski-type nonequilibrium work relation, $\overline{e^{-\beta W_i}}=e^{-\beta\Delta F^{eq}}$, and a corresponding nonequilibrium heat relation, $\overline{e^{-\beta Q'_i}}=1$, showing that the distributions of work, heat, and uncompensated heat are interdependent through entropy fluctuations. The approach provides microscopic insight into entropy production and connects stochastic fluctuations to classical thermodynamic quantities, with potential experimental relevance for probing energetic and entropic changes in driven small systems.

Abstract

The canonical distribution of Gibbs is extended to the case of systems outside equilibrium. The distribution of probabilities of a discrete energy levels system is used to provide a microscopic definition of work, along with a microscopic definition of the uncompensated heat of Clausius involved in nonequilibrium processes. The later is related to the presence of non-conservatives forces with regards to the variation of the external parameters. This new framework is used to investigate the nonequilibrium relations in stochastic thermodynamics. A new relation is derived for the random quantity of heat associated to the nonequilibrium work protocol. We finally show that the distributions of probabilities of work, heat and uncompensated heat are non-independent each other during a nonequilibrium process.

Figures (5)

• Figure 1: Fig. 1 Consider a thermodynamic system consisting of a closed volume containing particles, atoms or molecules, in perfect thermal contact with a heat bath maintained at a constant temperature $T_{0}$. The system is also coupled to a work reservoir, represented by a movable piston, which can perform or extract mechanical work. When work is supplied to the system with change of the external parameter $\lambda$, an amount of heat is transfered isothermally to the bath.
• Figure 2: Transformations between equilibrium states $\{A\}$ and $\{B\}$ in the $(\lambda,\xi)$ plane. The unique reversible (equilibrium) path is shown as a solid black line. Nonequilibrium trajectories are indicated by solid black lines with hatching, emphasizing that these trajectories are not uniquely determined. One representative nonequilibrium trajectory consists of two successive steps: first, step 1 at constant $\xi$, followed by step 2 at constant $\lambda$.
• Figure 3: Evolution along the two-step path, illustrated qualitatively in terms of effective energy levels for a simplified three-level system. The equilibrium energy levels of states $\{A\}$ and $\{B\}$ are shown as solid black lines. The effective energy levels of the intermediate nonequilibrium state $\{M\}$ are depicted by dashed black lines. Relaxation of these effective levels toward the equilibrium levels of state $\{B\}$ is indicated by a dashed black arrow.
• Figure 4: The distributions of work and uncompensated heat of Clausius are represented around their mean values (dot lines) in the same graph. The abscisse represents energy in arbitrary units of Joule. See text for explanation.
• Figure 5: The distributions of work and uncompensated heat of Clausius, as well as that of heat, are shown in the same graph, each spread around their respective mean values. The abscisse represents energy in arbitrary units of Joule. See text for details.