A variational formulation of stochastic thermodynamics: Spatially extended systems

TL;DR

This paper develops a variational framework for stochastic field theories that ensures thermodynamic consistency and local detailed balance, extending standard stochastic thermodynamics to spatially extended systems.

Contribution

It introduces a Hamiltonian-based variational formulation that guarantees thermodynamic consistency and derives natural fluctuation-dissipation relations for stochastic fields.

Findings

Entropy production form matches standard stochastic thermodynamics.

Emergence of fluctuation-dissipation relations from the variational principle.

Framework applies to complex fluids and structure-preserving numerical schemes.

Abstract

Stochastic field theories are often constructed phenomenologically, without a systematic assessment of thermodynamic consistency or local detailed balance. This may hinder a physical description of irreversibility at the field-theoretic level beyond the standard statistical formulation of stochastic thermodynamics. Here, we develop a variational formulation for thermodynamically consistent stochastic field theories by extending Hamilton's principle of classical field theory. Introducing the second law as an axiom yields a consistent local thermodynamic structure in which novel fluctuationdissipation relations emerge naturally, ensuring local detailed balance. Remarkably, the resulting entropy production takes the same form as in standard stochastic thermodynamics, up to a reformulation in an extended phase space incorporating both configurational and thermal variables. This correspondence extends key results, including individual trajectory-level thermodynamics and fluctuation theorems. The construction is formulated within a unified geometric framework based on a generalized Lagrange-d'Alembert principle, providing a top-down connection between phenomenological modeling and thermodynamic consistency. Potential applications include thermodynamically consistent modeling of complex fluids, Lagrangian reduction by symmetry in continuum systems, and structure-preserving numerical schemes for stochastic partial differential equations.