Weak invariants in dissipative systems: Action principle and Noether charge for kinetic theory

TL;DR

This paper explores the concept of weak invariants in classical dissipative systems, revealing their role as Noether charges within an action principle framework for kinetic equations like the Fokker-Planck equation, thus linking symmetry principles to nonequilibrium thermodynamics.

Contribution

It introduces a novel application of weak invariants as Noether charges in classical kinetic theory, connecting symmetry principles with dissipative system descriptions.

Findings

The auxiliary field in the kinetic action is a weak invariant.

The auxiliary field acts as a Noether charge.

The action remains invariant under transformations generated by the weak invariant.

Abstract

In nonequilibrium classical thermostatistics, the state of a system may be described by not only dynamical/thermodynamical variables but also a kinetic distribution function. This "double structure" bears some analogy with that in quantum thermodynamics, where both dynamical variables and the Hilbert space are involved. Recently, the concept of weak invariants has repeatedly been discussed in the context of quantum thermodynamics. A weak invariant is defined in such a way that its value changes in time but its expectation value is conserved under time evolution prescribed by a kinetic equation. Here, a new aspect of a weak invariant is revealed for the classical Fokker-Planck equation as an example of classical kinetic equations. The auxiliary field formalism is applied to construction of the action for the kinetic equation. Then, it is shown that the auxiliary field is a weak invariant and is the Noether charge. The action is invariant under the transformation generated by the weak invariant. The result may shed light on possible roles of the symmetry principle in the kinetic descriptions of nonequilibrium systems.