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Quantum current algebra symmetry and description of Boltz-mann type kinetic equations in statistical physics
* 1 , 2 , 3
1  Department of Physics, Mathematics and Computer Science at the Cracov University of Technology, Krakow, Poland
2  The Department of Computer Science and Teleautomatics at the Cracov University of Technology, Krakow, Poland
3  The Department of Mathematics at the Cracow Agriculture University, Cracow, Poland
Academic Editor: Eduardo Guendelman (registering DOI)

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 re‡ection 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