Extending Hamilton's principle with the second law axiom produces thermodynamically consistent stochastic field theories featuring natural fluctuation-dissipation relations and standard entropy production in extended phase space.
Graham, Covariant formulation of non-equilibrium statistical thermodynamics, Zeitschrift f¨ ur Physik B Condensed Matter and Quanta26, 397 (1977)
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A variational formulation of stochastic thermodynamics: Spatially extended systems
Extending Hamilton's principle with the second law axiom produces thermodynamically consistent stochastic field theories featuring natural fluctuation-dissipation relations and standard entropy production in extended phase space.