A variational formulation of stochastic thermodynamics. Part I: Finite-dimensional systems

TL;DR

This work develops a variational foundation for stochastic thermodynamics of finite-dimensional, continuous-time systems by enlarging the phase space to include thermodynamic entropy and introducing nonlinear nonholonomic constraints, so that the second law $\ abla S_{tot} \\ge 0$ enforces a thermodynamically consistent structure. The approach yields generalized fluctuation–dissipation relations and a trajectory-level description that aligns with standard ST, connecting to KL-based entropy production and integral fluctuation theorems, and extends naturally to both closed and open systems with state-dependent parameters and nonlinear couplings. Through three finite-dimensional examples, including a thermomechanical isolated system, an interconnected reservoir setup, and a thermomechanical open system, the framework recovers equilibrium distributions, Onsager reciprocity, and master fluctuation theorems, while accommodating cross-correlated noise and entropy pumping. Overall, the variational ST formalism provides a unifying, modular, and thermodynamically consistent methodology for modeling stochastic systems, with potential applications to active and complex fluids and to continuum field theories via future extensions.

Abstract

In this paper, we develop a variational foundation for stochastic thermodynamics of finite-dimensional, continuous-time systems. Requiring the second law (non-negative average total entropy production) systematically yields a consistent thermodynamic structure, from which novel generalized fluctuation-dissipation relations emerge naturally, ensuring local detailed balance. This principle extends key results of stochastic thermodynamics including an individual trajectory level description of both configurational and thermal variables and fluctuation theorems in an extended thermodynamic phase space. It applies to both closed and open systems, while accommodating state-dependent parameters, nonlinear couplings between configurational and thermal degrees of freedom, and cross-correlated noise consistent with Onsager symmetry. This is achieved by establishing a unified geometric framework in which stochastic thermodynamics emerges from a generalized Lagrange-d'Alembert principle, building on the variational structure introduced by Gay-Balmaz and Yoshimura [Phil. Trans. R. Soc. A 381, 2256 (2023)]. Irreversible and stochastic forces are incorporated through nonlinear nonholonomic constraints, with entropy treated as an independent dynamical variable. This work provides a novel approach for thermodynamically consistent modeling of stochastic systems, and paves the way to applications in continuum systems such as active and complex fluids.

Figures (3)

• Figure 1: The system $\Omega := \{(\bm{\omega}, s)\}$ is closed under internal irreversible processes $J_\alpha$, which involve information loss into the subspace of hidden DoFs represented by the thermodynamic entropy $s$. The system becomes open when subject to external fluxes, exchanging energy and entropy via work $\dot{w}$ and additional irreversible processes $J_\beta$.
• Figure 2: Schematic of an interconnected system with two reservoirs exchanging heat ($J$) and mass ($\mathcal{J}$). Cross-effects induced by cross-correlated noise lead to additional fluxes $\mathcal{J}_{cross}$, enabling heat (mass) transfer in response to chemical potential (temperature) gradients.
• Figure 3: Schematic of a thermomechanical open system. (1) denotes external heat flux, (2) represents entropy pumping (associated with work exchange), and (3) indicates internal dissipation. The entropy flow $\dot{s}_f$ captures contributions external to the system $\Omega$, corresponding to mechanisms (1) and (2). The thermodynamic entropy $s$ accounts for heat dissipated via processes (1) and (3). In contrast, the medium EPR $\dot{\Sigma}$ includes all contributions from (1), (2), and (3). Together, $\dot{\Sigma}$ and the system EPR $\dot \mathcal{S}$ characterize the total entropy production in the super-system comprising $\Omega$, the thermal reservoir $T^h$, and the external agent.