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arXiv:1009.1814 · Kinetic and Related Models

Heisenberg picture of quantum kinetic evolution in mean-field limit

V. I. Gerasimenko

Published Jan 1, 2011• 16 citations
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Abstract

We develop a rigorous formalism for the description of the evolution of observablesof quantum systems of particles in the mean-field scaling limit. The correspondingasymptotics of a solution of the initial-value problem of the dual quantum BBGKYhierarchy is constructed. Moreover, links of the evolution of marginal observablesand the evolution of quantum states described in terms of a one-particle marginaldensity operator are established. Such approach gives the alternative descriptionof the kinetic evolution of quantum many-particle systems.

DOI
10.3934/krm.2011.4.385

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We develop a rigorous formalism for the description of the evolution of observablesof quantum systems of particles in the mean-field scaling limit. The correspondingasymptotics of a solution of the initial-value problem of the dual quantum BBGKYhierarchy is constructed. Moreover, links of the evolution of marginal observablesand the evolution of quantum states described in terms of a one-particle marginaldensity operator are established. Such approach gives the alternative descriptionof the kinetic evolution of quantum many-particle systems.
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.
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 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 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.
We review new approaches to the description of the evolution of states of large quantum particle systems by means of the marginal correlation operators.Using the definition 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 large quantum particle systems.Furthermore, we consider the problem of the rigorous description of collective behavior of quantum manyparticle 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.