arXiv subjects
Publications and source records attributed to V. I. Gerasimenko.
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 chapter presents some new approaches to describing the collective behavior of complex systems of mathematical biology based on the evolution equations of observables such as open systems. This representation of kinetic evolution has looked to be the most direct and mathematically fully consistent formulation modeling the collective behavior of biological systems since the traditionally used concept of the state in kinetic theory is more subtle and is an implicit characteristic of the populations of living creatures. One of the advantages of the developed approach is the opportunity to construct kinetic equations for open complex systems in scaling approximations, involving initial correlations, in particular, that can characterize the condensed states of active soft matter. An approach is also related to the challenge of a rigorous derivation of the non-Markovian kinetic equations from underlying many-entity dynamics, which makes it possible to describe the memory effects of the collective behavior of living creatures.
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 article provides an overview of some advances in the mathematical understanding of the nature of the kinetic equations of quantum systems of many particles. The fundamental equations of modern mathematical physics are studied, in particular, the hierarchies of evolution equations of quantum systems and their asymptotic behavior described by kinetic nonlinear equations.
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.
In this survey the possible approaches to the description of the evolution of states of quantum many-particle systems by means of the possible modifications of the density operator which kernel known as density matrix are considered. In addition, an approach to the description of the evolution of states by means of the state of a typical particle of a quantum system of many particles is discussed or in other words, the foundations of describing the evolution of states by kinetic equations are considered.
We consider a new approach to the description of the collective behavior of complex systems of mathematical biology based on the evolution equations for observables of such systems. This representation of the kinetic evolution seems, in fact, the direct mathematically fully consistent formulation modeling the collective behavior of biological systems since the traditional notion of the state in kinetic theory is more subtle and it is an implicit characteristic of the populations of living creatures.
The paper deals with the problem of the rigorous description of the evolution of states of large particle quantum systems by means of correlation operators. A nonperturbative solution of the Cauchy problem of the hierarchy of nonlinear evolution equations for a sequence of marginal correlation operators is constructed. For initial states specified in terms of a one-particle density operator and correlation operators we also develop an approach to the description of the processes of the creation and the propagation of correlations within the framework of a one-particle density operator. Moreover, the mean field asymptotic behavior of constructed marginal correlation operators is established.
We consider semigroups of operators for hierarchies of evolution equations of large particle systems, namely, of the dual BBGKY hierarchy for marginal observables and the BBGKY hierarchy for marginal distribution functions. We establish that the generating operators of the expansions for one-parametric families of operators of these hierarchies are the corresponding order cumulants (semi-invariants) of semigroups for the Liouville equations. We also apply constructed semigroups to the description of the kinetic evolution of interacting stochastic Markovian processes, modeling the microscopic evolution of soft active matter. For this purpose we consider the mean field asymptotic behavior of the semigroup generated by the dual BBGKY hierarchy for marginal observables. The constructed scaling limit is governed by the set of recurrence evolution equations, namely, by the Vlasov-type dual hierarchy. Moreover, the relationships of this hierarchy of evolution equations with the Vlasov-type kinetic equation with initial correlations are established.
In the paper we consider the problem of the rigorous description of the kinetic evolution in the presence of initial correlations of quantum large particle systems. One of the developed approaches consists in the description of the evolution of quantum many-particle systems within the framework of marginal observables in mean field scaling limit. Another method based on the possibility to describe the evolution of states within the framework of a one-particle marginal density operator governed by the generalized quantum kinetic equation in case of initial states specified by a one-particle marginal density operator and correlation operators.
We develop a rigorous formalism for the description of the kinetic evolution of many-particle systems with the dissipative interaction. The relationships of the evolution of a hard sphere system with inelastic collisions described within the framework of marginal observables governed by the dual 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.
We develop a rigorous formalism for the description of the kinetic evolution of interacting entities modeling systems in mathematical biology within the framework of the evolution of marginal observables. For this purpose we construct the mean field asymptotic behavior of a solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables of the dynamical systems based on the Markov jump processes, exhibiting the intrinsic properties of the living entities. The constructed scaling limit is governed by the set of recurrence evolution equations, namely by the dual Vlasov-type hierarchy. Moreover, the relationships of the dual Vlasov hierarchy for the limit marginal observables with the Vlasov-type kinetic equation is established.
In the paper the possible approaches to the rigorous derivation of the Boltzmann kinetic equation with hard sphere collisions from underlying dynamics are considered. In particular, a formalism for the description of the evolution of infinitely many hard spheres within the framework of marginal observables in the Boltzmann--Grad scaling limit is developed. Also we give consideration to one more approach of the description of the kinetic evolution of hard spheres in terms of a one-particle distribution function governed by the non-Markovian generalization of the Enskog equation and the Boltzmann--Grad asymptotic behavior of its non-perturbative solution is established.
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.