**Quantum current algebra symmetry and description of Boltz-mann type kinetic equations in statistical physics**

**Published:**07 August 2021 by

**MDPI**in

**Symmetry 2021 - The 3rd International Conference on Symmetry**session

**Physics and Symmetry**

**Abstract:**

We study a special class of dynamical systems of Boltzmann-Bogolubov and Boltzmann-

Vlasov type on in
nite dimensional functional manifolds modeling kinetic processes in many-

particle media. Based on algebraic properties of the canonical quantum symmetry current

algebra and its functional representations we proposed a new approach to invariant reducing

the Bogolubov hierarchy on a suitably chosen correlation function constraint and deducing

the related modi
ed Boltzmann-Bogolubov kinetic equations on a
nite set of multiparticle

distribution functions. It is well known that the classical Bogolubov-Boltzmann kinetic equations under the

condition of manyparticle correlations at weak short range interaction

potentials describe long waves in a dense gas medium. The same equation, called the Vlasov

one, as it was shown by N. Bogolubov, describes also exact microscopic solutions of the

in
nite Bogolubov chain or the manyparticle distribution functions, which was widely

studied making use of both classical approaches and making use

of the generating Bogolubov functional method and the related quantum current algebra

representations. In this case the Bogolubov equation for distribution functions in some domain. Remark here that the basic kinetic equation is reversible under the time reection t-->-t, thus it is obvious that it can not describe

thermodynamically stable limiting states of the particle system in contrast to the classical

Bogolubov-Boltzmann kinetic equations, being a priori time nonre-

versible owing to the choice of special boundary conditions. This

means that in spite of the Hamiltonicity of the Bogolubov chain for the distribution func-

tions, the Bogolubov-Boltzmann equation a priori is not reversible. The classical Poisson bracket expression allows a slightly diffeerent

Lie-algebraic interpretation, based on considering the functional space D( M_f_1 ) as a Pois-

sonian manifold, related with the canonical symplectic structure on the di¤eomorphism

group Diff() of the domain R^3,
first described still in 1887 by Sophus Lie.

These aspects and its di¤erent consequences are analyzed in detail in our report.

**Keywords:**kinetic equations; Bogolubov chain; multi-particle distribution functions; current algebra; Wigner representation; functional equations; reduction theory; Lie-Poisson bracckets; invariant functional submanifolds