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Solutions of Bogolyubov equations for infinite three-dimensional systems of particles

D. Ya. Petrina, V. I. Gerasimenko, P. V. Malyshev

De Gruyter eBooksDec 31, 19871 citation

Bogolyubov's method in the quantum theory of a crystal with many-particle interaction

Edward V. Gevorkyan

Theoretical and Mathematical PhysicsFeb 1, 19763 citations

On the Initial-Value Problem to the Quantum Dual BBGKY Hierarchy

G. Borgioli, V. I. Gerasimenko

We develop a rigorous formalism for the description of the evolution of observables in quantum systems of particles. We construct a solution of the initial-value problem to the quantum dual BBGKY hierarchy of equations as an expansion over particle clusters whose evolution are governed by the corresponding-order dual cumulant (dual semi-invariant) of the evolution operators of finitely many particles. For initial data from the space of sequences of bounded operators the existence and uniqueness theorem is proved.We develop a rigorous formalism for the description of the evolution of observables in quantum systems of particles. We construct a solution of the initial-value problem to the quantum dual BBGKY hierarchy of equations as an expansion over particle clusters whose evolution are governed by the corresponding-order dual cumulant (dual semi-invariant) of the evolution operators of finitely many particles. For initial data from the space of sequences of bounded operators the existence and uniqueness theorem is proved.

arXiv:0806.1027Jun 5, 200814 citations

Bogolyubov kinetic equation for quantum dynamical systems

Дмитрий Петрович Санкович

Soviet physics. DokladyOct 1, 19850 citations

Quantum Kinetic Equations and Evolution of Many-Particle Systems

V. I. Gerasimenko

In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.

arXiv:0908.2797Aug 19, 20090 citations

Dynamics of correlations of Bose and Fermi particles

V. I. Gerasimenko, Dmytro Polishchuk

We discuss the origin of the microscopic description of correlations in quantum many-particle systems obeying Fermi–Dirac and Bose–Einstein statistics. For correlation operators that give the alternative description of the quantum state evolution of Bose and Fermi particles, we deduce the von Neumann hierarchy of nonlinear equations and construct the solution of its initial-value problem in the corresponding spaces of sequences of trace class operators. The links of constructed solution both with the solution of the quantum BBGKY hierarchy and with the nonlinear BBGKY hierarchy for marginal correlation operators are discussed. The solutions of the Cauchy problems of these hierarchies are constructed, in particular for initial data satisfying a chaos property. Copyright © 2010 John Wiley & Sons, Ltd.We discuss the origin of the microscopic description of correlations in quantum many-particle systems obeying Fermi–Dirac and Bose–Einstein statistics. For correlation operators that give the alternative description of the quantum state evolution of Bose and Fermi particles, we deduce the von Neumann hierarchy of nonlinear equations and construct the solution of its initial-value problem in the corresponding spaces of sequences of trace class operators. The links of constructed solution both with the solution of the quantum BBGKY hierarchy and with the nonlinear BBGKY hierarchy for marginal correlation operators are discussed. The solutions of the Cauchy problems of these hierarchies are constructed, in particular for initial data satisfying a chaos property. Copyright © 2010 John Wiley & Sons, Ltd.

arXiv:1001.3893Mathematical Methods in the Applied SciencesJun 18, 201017 citations

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