Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2503.08412 · Methods of Functional Analysis and Topology

Cumulant Expansions of Groups of Operators for Particle Systems with Topological Nearest-neighbor Interaction

V. I. Gerasimenko, Igor V. Gapyak

Published Jan 1, 2025• 0 citations
arXivHTMLPDF

Abstract

This article discusses the structure of expansions that represent non-perturbative solutions of the Cauchy problem for the evolution equation hierarchies for the state and observables of many-particle systems with topological nearest-neighbor interaction. The generating operators for these expansions are derived using a proposed cluster expansion method applied to the groups of operators in the Liouville equations for both states and observables, respectively. The article also introduced the concept of a cumulant representation for distribution functions that describe the state of many particles with topological interactions and constructs a non-perturbative solution to the Cauchy problem for the hierarchy of nonlinear evolution equations for the cumulants of distribution functions. Furthermore, a relationship is established between the constructed solution and the series expansion structures for reduced distribution and correlation functions.

DOI
10.31392/mfat-npu26_1.2025.02

Related papers

Evolution Equations for Cumulants of Distribution Functions of Particle Systems with Topological Interaction

V. I. Gerasimenko, Igor V. Gapyak

The article formulates the concept of a cumulant representation for distribution functions that describe the state of many-particle systems with topological interaction, i.e., using the interaction potential determined by the proximity rank of particles. Cumulants of probability distribution functions are interpreted as correlations of particle states and are defined as solutions of the corresponding cluster expansions of probability distri­bution functions. We emphasize that the correlations that arise during the evolution of a system of particles with topological interaction naturally differ from the structure of correlations of many-particle systems, the state of which is traditionally described by symmetric probability distribution functions. The article establishes a hierarchy of evolution nonlinear equations for correlation functions (hierarchy of recursive Liouville equations). In the space sequences of integrated functions, a non-perturbative solution of the Cauchy problem for a hierarchy of such nonlinear evolution equations is constructed. Typical properties of the expansion of such a solution, which are generated by the properties of its generating operators, namely, cumulants of groups of operators of Liouville equations, are investigated. Based on the dynamics of correlations, the structure of series expansions is established, which determines reduced distribution functions, as well as reduced correlation functions, which, in particular, made it possible to substantiate the structure of generating operators of the solution of the Cauchy problem for the hierarchy of BBGKY equations (Bogolyubov–Born–Green–Kirkwood–Yvon). It is proved that the structure of expansions for correlation functions, the generating operators of which are cumulants of groups of operators of the corresponding order of Liouville equations, induces the cumulant structure of series expansions for reduced distribution functions and reduced correlation functions. Thus, the dynamics of state correlations generate the dynamics of many-particle systems with topological interaction.

Mathematical and computer modelling Series Physical and mathematical sciences•Dec 26, 2024•0 citations

Cumulant Expansions of the Operator Groups of Quantum Many-Particle Systems

Ihor Gapyak, Viktor Gerasimenko

UDC 530.145:517.98 We present a method of cluster expansions for groups of operators associated with the von-Neumann equations for states and the Heisenberg equations for observables aimed at construction of generating operators for the solutions to the Cauchy problem for hierarchies of evolution equations of many-particle quantum systems.

arXiv:2512.05036•Ukrainian Mathematical Journal•Mar 28, 2026•0 citations

ЕВОЛЮЦІЯ РІВНЯНЬ ДЛЯ МАРГІНАЛЬНИХ ОПЕРАТОРІВ КОРЕЛЯЦІЇ

Gerasimenko, Victor Ivanovich, Polishchuk, Denis Olegovich

This paper is devoted to the problem of the description of nonequilibrium correlations of quantum many-particle systems. A non-perturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed as an expansion over particle clusters which evolution is governed by the corresponding-order cumulant of the nonlinear groups of operators generated by the von Neumann hierarchy.

The Scientific Issues of Ternopil Volodymyr Hnatiuk National Pedagogical University Series pedagogy•May 6, 2011•0 citations

Evolution of correlations of quantum many-particle systems

V. I. Gerasimenko, V. O. Shtyk

We consider the origin of the microscopic description of correlations in quantum systems. For correlation operators that give the alternative description of the quantum state evolution of many-particle systems, we deduce the von Neumann hierarchy of nonlinear equations and discuss the relationships of its solution both with the solution of the BBGKY hierarchy and with the s -particle correlation operators. A solution of the von Neumann hierarchy 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 of the von Neumann equations for the density operators. 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.

arXiv:0712.4336•Journal of Statistical Mechanics Theory and Experiment•Mar 12, 2008•14 citations

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

V. I. Gerasimenko

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.

arXiv:0804.1153•Birkhäuser Basel eBooks•Jan 1, 2009•5 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.

arXiv:0806.1027•Jun 5, 2008•14 citations
This article discusses the structure of expansions that represent non-perturbative solutions of the Cauchy problem for the evolution equation hierarchies for the state and observables of many-particle systems with topological nearest-neighbor interaction. The generating operators for these expansions are derived using a proposed cluster expansion method applied to the groups of operators in the Liouville equations for both states and observables, respectively. The article also introduced the concept of a cumulant representation for distribution functions that describe the state of many particles with topological interactions and constructs a non-perturbative solution to the Cauchy problem for the hierarchy of nonlinear evolution equations for the cumulants of distribution functions. Furthermore, a relationship is established between the constructed solution and the series expansion structures for reduced distribution and correlation functions.
The article formulates the concept of a cumulant representation for distribution functions that describe the state of many-particle systems with topological interaction, i.e., using the interaction potential determined by the proximity rank of particles. Cumulants of probability distribution functions are interpreted as correlations of particle states and are defined as solutions of the corresponding cluster expansions of probability distri­bution functions. We emphasize that the correlations that arise during the evolution of a system of particles with topological interaction naturally differ from the structure of correlations of many-particle systems, the state of which is traditionally described by symmetric probability distribution functions. The article establishes a hierarchy of evolution nonlinear equations for correlation functions (hierarchy of recursive Liouville equations). In the space sequences of integrated functions, a non-perturbative solution of the Cauchy problem for a hierarchy of such nonlinear evolution equations is constructed. Typical properties of the expansion of such a solution, which are generated by the properties of its generating operators, namely, cumulants of groups of operators of Liouville equations, are investigated. Based on the dynamics of correlations, the structure of series expansions is established, which determines reduced distribution functions, as well as reduced correlation functions, which, in particular, made it possible to substantiate the structure of generating operators of the solution of the Cauchy problem for the hierarchy of BBGKY equations (Bogolyubov–Born–Green–Kirkwood–Yvon). It is proved that the structure of expansions for correlation functions, the generating operators of which are cumulants of groups of operators of the corresponding order of Liouville equations, induces the cumulant structure of series expansions for reduced distribution functions and reduced correlation functions. Thus, the dynamics of state correlations generate the dynamics of many-particle systems with topological interaction.
We consider the origin of the microscopic description of correlations in quantum systems. For correlation operators that give the alternative description of the quantum state evolution of many-particle systems, we deduce the von Neumann hierarchy of nonlinear equations and discuss the relationships of its solution both with the solution of the BBGKY hierarchy and with the s -particle correlation operators. A solution of the von Neumann hierarchy 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 of the von Neumann equations for the density operators. 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.
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.
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.