Noether's theorem applied to GENERIC

TL;DR

This work extends the Noether-charge interpretation of thermodynamic entropy to macroscopic dynamics described by PRE-GENERIC and GENERIC under quasistatic control. By embedding pre-GENERIC into dynamical fluctuation theory and constructing explicit Lagrangian/Hamiltonian formalisms, the authors identify a continuous symmetry of the path-space action whose Noether charge is the thermodynamic entropy, valid in both reversible (Hamiltonian) and dissipative (gradient) components. The analysis is illustrated with a nonlinear friction example, showing that, in the adiabatic limit, entropy remains invariant along quasistatic trajectories, while for open systems the free energy replaces entropy as the relevant Noether charge. The results bridge Clausius’ Heat Theorem and Noether’s theorem, offering a unified variational perspective on entropy conservation in macroscopic non-equilibrium dynamics. This framework clarifies when entropy is conserved (closed, adiabatic) versus when free energy governs the symmetry, and provides a concrete path-space formalism for calculating fluctuations and conserved quantities in GENERIC-type systems.

Abstract

The last decades have seen growing interest in connecting principles of thermodynamics with methods from analytical mechanics. The thermodynamic formalism has become an inspiring framework in the study of smooth dynamical systems, and pioneering works of Helmholtz, Clausius, and Boltzmann have been reinstated as possible dynamical foundations of the (first part of the) Heat Theorem. The present paper follows the work of Wald et al., where black hole entropy was identified as a Noether charge. The adiabatic invariance of the thermodynamic entropy indeed invites a connection with Noether's theorem, and has been the subject of various papers. Here we add the case of GENERIC, a macroscopic dynamics whose acronym stands for ``General Equation for Non-Equilibrium Reversible-Irreversible Coupling''. Its evolution has two contributions: a dissipative part, which is of a generalized gradient descent form, and a Hamiltonian flow. We consider a quasistatic protocol for external parameters, and we embed GENERIC as the zero-cost flow for a Lagrangian governing the dynamical fluctuations. We find a continuous symmetry of the corresponding path-space action with the thermodynamic entropy as Noether charge, both in the Lagrangian and Hamiltonian formalisms. We make the calculations explicit through the example of an inertial probe with nonlinear friction.