Abstract
We review a non-relativistic current algebra symmetry approach to constructing the Bogolubov generating functional of many-particle distribution functions and apply it to description of invariantly reduced Hamiltonian systems of the Boltzmann type kinetic equations, related to naturally imposed constraints on many-particle correlation functions. As an interesting example of deriving Vlasov type kinetic equations, we considered a quantum-mechanical model of spinless particles with delta-type interaction, having applications for describing so called Benney-type hydrodynamical praticle flows. We also review new results on a special class of dynamical systems of Boltzmann–Bogolubov and Boltzmann–Vlasov type on infinite 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 succeeded in dual analysis of the infinite Bogolubov hierarchy of many-particle distribution functions and their Hamiltonian structure. Moreover, we proposed a new approach to invariant reduction of the Bogolubov hierarchy on a suitably chosen correlation function constraint and deduction of the related modified Boltzmann–Bogolubov kinetic equations on a finite set of multi-particle distribution functions. There are also presented results of application of devised methods to describing kinetic properties of a many-particle system with an adsorbent surface, in particular, the corresponding kinetic equation for the occupation density distribution function is derived.
Highlights
As it is well known [1,2,3,4,5,6,7,8], the objective of the kinetic theory of many-particle dynamical systems is to explain the properties of macroscopic non-equilibrium state of gases in terms of microscopic properties of individual gas molecules and interaction forces between them
He has found “microscopic” expressions for transport coefficients for gas constituted by the so-called Maxwell molecules, which interact through push forces, which are inversely proportional to the interparticle distance in the fifth power. He has shown that the viscosity and thermal conductivity coefficients do not depend on the gas density. Another significant contribution to the statistical mechanics theory was made by Boltzmann, who obtained his world-known equation for time-dependent distribution function for dilute gas in nonequilibrium state in 1872
We study in detail a special class of dynamical systems of Boltzmann–Bogolubov and Boltzmann–Vlasov type on infinite dimensional functional manifolds modeling kinetic processes in many-particle media
Summary
As it is well known [1,2,3,4,5,6,7,8], the objective of the kinetic theory of many-particle dynamical systems is to explain the properties of macroscopic non-equilibrium state of gases (or liquids) in terms of microscopic properties of individual gas molecules and interaction forces between them. He has found “microscopic” expressions for transport coefficients for gas constituted by the so-called Maxwell molecules, which interact through push forces, which are inversely proportional to the interparticle distance in the fifth power He has shown that the viscosity and thermal conductivity coefficients do not depend on the gas density. It was found that the hierarchy of the Bogolubov equations in terms of the quantum generating functional realizes irreducible unitary representations of the quantum current Lie algebra G = Diff(Λ) J (Λ;R), which is a semi-direct product of the group of diffeomorphisms of a domain Λ ⊂ R3 and the space of Schwartz functions on it Within this approach all classic solutions are obtained, respectively, after moving to Wigner representations [3,10,13,14] for basic operators of the initial algebraic structure. There are presented results of application of devised methods to describing kinetic properties of a many-particle system with an adsorbent surface, in particular, there is derived the corresponding kinetic equation for the occupation density distribution function
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