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

Abstract

<jats:p>In non-equilibrium 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 the 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 non-equilibrium systems.</jats:p> <jats:p>This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.</jats:p>

Quantum Weak Invariants: Dynamical Evolution of Fluctuations and Correlations

“…In recent work [ 26 ], the action principle has been formulated for a quantum master equation based on the auxiliary field formalism [ 27 ], and it has been found that the auxiliary field is actually a weak invariant. In addition, the auxiliary field has also shown to be the Noether charge [ 28 ]. This reminds one of another observation: the black hole entropy related to surface gravity can be regarded as the Noether charge [ 29 ].…”

Nonequilibrium thermodynamics: emergent and fundamental

“…The third benchmark, backward compatibility, should precede the mathematical structure. The compatibility with a statistical treatment, based on a given microscopic or mesoscopic material composition, is a clear benchmark of any general phenomenology and appears here in the papers [1,10,12]. In particular, the compatibility with the kinetic theory is important in [3,[6][7][8].…”

“…The usual approach attempts to extend the validity of variational principles to dissipative evolution [13]. In [12], we see a promising new method. On the other hand, GENERIC (General Equation for the Nonequilibrium Reversible-Irreversible Coupling) treats the dissipative-nondissipative parts on equal footing [3,7].…”

Variational principles and nonequilibrium thermodynamics

“…are not symmetric; therefore, special modifications are necessary to construct a variational principle in order to circumvent the strict mathematical conditions [3][4][5]. The applied tricks can be very different: for instance, changing the original equations, increasing the number of variables, reducing the corresponding function spaces, turning to statistical interpretations, and so on [6][7][8][9][10][11][12][13]. The obtained variational principles are special, different and not equivalent [14].…”

“…The applied tricks can be very different: for instance, changing the original equations, increasing the number of variables, reducing the corresponding function spaces, turning to statistical interpretations, and so on [6][7][8][9][10][11][12][13]. The obtained variational principles are special, different and not equivalent [14].…”