Noether's theorem applied to GENERIC

TL;DR

This paper extends Noether's theorem to GENERIC, a framework for non-equilibrium thermodynamics, revealing a symmetry where thermodynamic entropy acts as a conserved charge during quasistatic processes.

Contribution

It demonstrates a continuous symmetry in GENERIC dynamics with thermodynamic entropy as a Noether charge, linking thermodynamics and analytical mechanics.

Findings

Identifies a Noether symmetry in GENERIC for quasistatic protocols.

Shows thermodynamic entropy as a conserved charge in the symmetry.

Provides explicit calculations with a nonlinear friction example.

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.