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PAMM

The Bogolubov generating functional method in statistical physics and the Wigner transform (Poster Presentation)

Anatoliy K. Prykarpatsky, Nikolai N. Bogolubov

Published Dec 1, 2007• 0 citations
Publisher

Abstract

Abstract We show that the N.N. Bogolubov generating functional method is a very effective tool for studying distribution functions of both equilibrium and non equilibrium states of classical many‐particle dynamical systems. In some cases the Bogolubov generating functionals can be represented by means of infinite Ursell‐Mayer diagram expansions, whose convergence holds under some additional constraints on statistical system. The classical Bogolubov idea [1] to use the Wigner density operator transformation for studying the non equilibrium distribution functions is developed and new analytic non‐stationary solution to the classical N.N. Bogolubov evolution functional equation is constructed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pamm.200700983

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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.
Abstract The new method for solving Wigner–Liouville's type equations and studying dynamics of quantum particles has been developed within the Wigner formulation of quantum statistical mechanics. This approach combines both molecular dynamics and Monte Carlo methods and computes traces and spectra of the relevant dynamical quantities. Considering, as an application, the quantum dynamics of an ensemble of interacting electrons in an array of random scatterers clearly demonstrates that the many‐particle interaction between the electrons can lead to an enhancement of the electrical conductivity. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Wigner representation for distribution functions is very important in modern physical theory: it allows the formulation of quantum mechanics and statistics in terms of classical theory. Under this approach, physical values are easily represented by appropriate Weyl transformations. It is demonstrated how the Wigner representation can be used both in equilibrium and non-equilibrium cases. Using this approach we demonstrate the equivalence between the Kadanoff-Baym and the Keldysh techniques in the non-equilibrium statistical mechanics.