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In nonequilibrium classical thermostatistics, the state of a system may bedescribed by not only dynamical/thermodynamical variables but also a kineticdistribution function. This "double structure" bears some analogy with that inquantum thermodynamics, where both dynamical variables and the Hilbert spaceare involved. Recently, the concept of weak invariants has repeatedly beendiscussed in the context of quantum thermodynamics. A weak invariant is definedin such a way that its value changes in time but its expectation value isconserved under time evolution prescribed by a kinetic equation. Here, a newaspect of a weak invariant is revealed for the classical Fokker-Planck equationas an example of classical kinetic equations. The auxiliary field formalism isapplied to construction of the action for the kinetic equation. Then, it isshown that the auxiliary field is a weak invariant and is the Noether charge.The action is invariant under the transformation generated by the weakinvariant. The result may shed light on possible roles of the symmetryprinciple in the kinetic descriptions of nonequilibrium systems.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

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