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arXiv:2610.28846

Methods for Deriving Nonlinear Kinetic Equations from the Dynamics of Quantum Systems

V. I. Gerasimenko

math-phcond-mat.stat-mechquant-phSubmitted Oct 20, 2026• Announced Oct 22, 2026• 0 citations
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Abstract

The article outlines key topics related to deriving kinetic equations and, in general, to the relationship between the evolution equations governing quantum many-particle systems. The main issues being considered are the mathematical origin of quantum kinetic equations, novel methods for deriving kinetic equations from hierarchies of evolution equations for quantum systems, specifically focusing on kinetics within the framework of the evolution of observables, and the generalization of the quantum kinetic equation specified by non-separable initial states.

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35 pages

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The article outlines key topics related to deriving kinetic equations and, in general, to the relationship between the evolution equations governing quantum many-particle systems. The main issues being considered are the mathematical origin of quantum kinetic equations, novel methods for deriving kinetic equations from hierarchies of evolution equations for quantum systems, specifically focusing on kinetics within the framework of the evolution of observables, and the generalization of the quantum kinetic equation specified by non-separable initial states.
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.
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 review some new approaches to the description of the evolution of states of many-particle quantum systems by means of the correlation operators. Using the denition of marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy, we establish that a sequence of such operators is governed by the nonlinear quantum BBGKY hierarchy. The constructed nonperturbative solution of the Cauchy problem to this hierarchy of nonlinear evolution equations describes the processes of the creation and the propagation of correlations in many-particle quantum systems. Moreover, we consider the problem of the rigorous description of collective behavior of many-particle quantum systems by means of a one-particle (marginal) correlation operator that is a solution of the generalized quantum kinetic equation with initial correlations, in particular, correlations characterizing the condensed states of systems.
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.