V. I. Gerasimenko

On the structure of expansions for the BBGKY hierarchy solutions

We consider classical many-particle systems of identical particles and distinguishable particles. For these types of systems we construct a new representation of a solution to the initial value problem to the BBGKY hierarchy of equations, namely, in the form of an expansion over particle clusters whose evolution is governed by the cumulants (semi-invariants) of the evolution operator of the corresponding particle cluster. Such a representation of solutions enables us to describe the cluster nature of the evolution of infinite particle systems with different symmetry properties in detail. A convergence of the constructed expansions is investigated in the suitable functional spaces.

Hard sphere dynamics and the Enskog equation

We develop a rigorous formalism for the description of the kinetic evolution of infinitely many hard spheres. On the basis of the kinetic cluster expansions of cumulants of groups of operators of finitely many hard spheres the nonlinear kinetic Enskog equation and its generalizations are justified. It is established that for initial states which are specified in terms of one-particle distribution functions the description of the evolution by the Cauchy problem of the BBGKY hierarchy and by the Cauchy problem of the generalized Enskog kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation are an equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.

Dynamics of correlations of Bose and Fermi particles

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 for initial data satisfying a chaos property.

Evolution of correlations of quantum many-particle systems

The Cauchy problem for the von Neumann hierarchy of nonlinear equations is investigated. One describes the evolution of all possible states of quantum many-particle systems by the correlation operators. A solution of such nonlinear equations is constructed in the form of an expansion over particle clusters whose evolution is described by the corresponding order cumulant (semi-invariant) of evolution operators for the von Neumann equations. For the initial data from the space of sequences of trace class operators the existence of a strong and a weak solution of the Cauchy problem is proved. We discuss the relationships of this solution both with the sparticle statistical operators, which are solutions of the BBGKY hierarchy, and with the s-particle correlation operators of quantum systems.

On quantum kinetic equations of many-particle systems in condensed states

We develop a rigorous formalism for the description of the evolution of states of quantum many-particle systems in terms of a one-particle density operator, which enables us to construct the kinetic equations in scaling limits in the presence of correlations of particle states at initial time, for instance, correlations characterizing the condensed states.

A description of the evolution of quantum states by means of the kinetic equation

We develop a rigorous formalism for the description of the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and by the Cauchy problem of the generalized quantum kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is established in the space of trace class operators. The links of the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed.We develop a rigorous formalism for describing the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator, the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum Bogolyubov–Born–Green–Kirkwood–Yvon hierarchy, and by the Cauchy problem of the generalized quantum kinetic equation, together with a sequence of explicitly defined functionals of a solution of a stated kinetic equation, is established in the space of trace-class operators. The links between the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed.

A nonperturbative solution of the nonlinear BBGKY hierarchy for marginal correlation operators

This paper is devoted to the problem of the description of nonequilibrium correlations in quantum many-particle systems. The nonlinear quantum BBGKY hierarchy for marginal correlation operators is rigorously derived from the von Neumann hierarchy for correlation operators that give an alternative approach to the description of states in comparison with the density operators. A nonperturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed.

Groups of Operators for Evolution Equations of Quantum Many-particle Systems

The aim of this work is to study the properties of groups of operators for evolution equations of quantum many-particle systems, namely, the von Neumann hierarchy for correlation operators, the BBGKY hierarchy for marginal density operators and the dual BBGKY hierarchy for marginal observables. We show that the concept of cumulants (semi-invariants) of groups of operators for the von Neumann equations forms the basis of the expansions for one-parametric families of operators of various evolution equations for infinitely many particles.

On the non-Markovian Enskog equation for granular gases

We develop a rigorous formalism for the description of the kinetic evolution of many-particle systems with dissipative interaction. The links of the evolution of a hard sphere system with inelastic collisions described within the framework of marginal observables governed by the dual Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy and the evolution of states described by the Cauchy problem of the Enskog kinetic equation for granular gases are established. Moreover, we consider the Boltzmann–Grad asymptotic behavior of the constructed non-Markovian Enskog kinetic equation for granular gases in a one-dimensional space.

Thermodynamic limit of nonequilibrium states of a three-dimensional system of elastic spheres

The authors study the evolution of a three-dimensional system of an infinite number of particles interacting as absolutely elastic spheres. The state of such a system is described by an infinite sequence of distribution functions, which satisfy the Cauchy problem for the Bogolyubov equations. A rigorous justification is given of the thermodynamic limit for the nonequilibrium states of a system of elastic spheres of arbitrairily many dimensions. The authors show that the limit distribution functions are weak solutions to the Cauchy problem for the Bogolyubov equations of infinite systems.