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Variational formulation of stochastic thermodynamics: Spatially extended systems

Héctor Vaquero del Pino1,*, François Gay-Balmaz2,†, Hiroaki Yoshimura3, and Lock Yue Chew1,‡

  • *Contact author: hector001@e.ntu.edu.sg
  • †Contact author: francois.gb@ntu.edu.sg
  • ‡Contact author: lockyue@ntu.edu.sg
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Phys. Rev. E 114, 044125 – Published 15 October, 2026

DOI: https://doi.org/10.1103/dd9y-9jc4

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 fluctuation-dissipation 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.

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Phys. Rev. E 114, 044125– Published 15 October, 2026
  • Received 24 April 2026
  • Revised 30 July 2026
  • Accepted 17 September 2026
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