arXiv subjects
Publications and source records attributed to I. V. Gapyak.
The article presents a method of cluster expansions for groups of operators associated with the von Neumann equations for states and the Heisenberg equations for observables, aiming to construct generating operators for nonperturbative solutions to the Cauchy problem for hierarchies of evolution equations of many-particle quantum systems.
The article presents the concept of a cumulant representation for distribution functions describing the states of many-particle systems with topological nearest-neighbor interaction. A solution to the Cauchy problem for the hierarchy of nonlinear evolution equations for the cumulants of distribution functions of such systems is constructed. The connection between the constructed solution and the series expansion structure for a solution to the Cauchy problem of the BBGKY hierarchy has been established. Furthermore, the expansion structure for a solution to the Cauchy problem of the hierarchy of evolution equations for reduced observables of topologically interacting particles is established.
The article deals with the challenge of the construction of solutions to hierarchies of fundamental evolution equations for many colliding particles. The method of cluster expansions of the groups of operators of the Liouville equations for observables and a state is used to establish the generating operators of expansions representing solutions of the Cauchy problems of the BBGKY hierarchy (Bogolyubov-Born-Green-Kirkwood-Yvon) as well as of the dual BBGKY hierarchy, respectively.
The review presents rigorous results of the theory of fundamental equations of evolution of many-particle systems with collisions and also considers their connection with nonlinear kinetic equations describing the collective behavior of particles in scaling approximations. This work is dedicated to the 160th anniversary of the birth of Dmytro Oleksandrovych Grave, the first academician of the Ukraine Academy of Sciences in mathematics and the founder of the Institute of Mathematics in 1920.
The paper develops an approach to the description of the evolution of correlations for many hard spheres based on a hierarchy of evolution equations for the cumulants of the probability distribution function governed by the Liouville equation. It is established that the constructed dynamics of correlations underlies the description of the evolution of the states of many hard spheres described by the BBGKY hierarchy for reduced distribution functions or the hierarchy of nonlinear evolution equations for reduced correlation functions. As an application of the developed approach to describing the evolution of the state of many hard spheres within the framework of dynamics of correlations, the challenges of the derivation of kinetic equations are discussed.
The possible ways to describe the states of a system of many hard spheres are considered, in particular by means of functions describing correlations of states. It is stated an approach to the description of the evolution based on the dynamics of correlations in a system of hard spheres. In addition, we consider another approach to describing the evolution of correlations in a system of many hard spheres, namely, in the framework of a one-particle distribution function (correlation function) governed by the non-Markovian Enskog kinetic equation.
The article discusses some of the latest advances in the mathematical understanding of the nature of kinetic equations that describe the collective behavior of many-particle systems with collisional dynamics.
The rigorous approach to the description of the kinetic evolution of a many-particle system composed of a trace hard sphere and an environment of finitely many hard spheres is developed. We prove that the evolution of states of a trace hard sphere in an environment can be described within the framework of the marginal distribution function governed by the generalized Fokker -- Planck kinetic equation and an infinite sequence of the explicitly defined functionals of this function.
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 is an equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.