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Theoretical and Mathematical Physics

Evolution of a quantum system of many particles interacting via the generalized Yukawa potential

Nikolai Nikolaevich Bogoliubov, M. Yu. Rasulova, Umarbek Avazov

Published Dec 1, 2016• 2 citations
Publisher

Abstract

We study the evolution of a system of N particles that have identical masses and charges and interact via the generalized Yukawa potential. The system is placed in a bounded region. The evolution of such a system is described by the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) chain of quantum kinetic equations. Using semigroup theory, we prove the existence of a unique solution of the BBGKY chain of quantum kinetic equations with the generalized Yukawa potential.

DOI
10.1134/s0040577916120102

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We study the evolution of a system of N particles that have identical masses and charges and interact via the generalized Yukawa potential. The system is placed in a bounded region. The evolution of such a system is described by the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) chain of quantum kinetic equations. Using semigroup theory, we prove the existence of a unique solution of the BBGKY chain of quantum kinetic equations with the generalized Yukawa potential.
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing an approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in the case of an adiabatically slow external perturbation, though it is not restricted to adiabatic systems. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of particles with cubic repulsion. Equilibrium statistical mechanics of this Pechukas gas allows us to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the Pechukas gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e., approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and interlevel transition amplitudes, they allow us to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N1/N1/N, where NNN is the number of the ``Pechukas gas particles'' (i.e., energy levels in the system).
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case of an adiabatically slow external perturbation. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of with cubic repulsion. Equilibrium statistical mechanics of this gas allows to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e. approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and inter-level transition amplitudes, they allow to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N, where N is the number of the Pechukas gas particles (i.e. energy levels in the system).
We consider the problem of the rigorous description of nonequilibrium quantum correlations. Within the framework of an alternative approach to the description of the evolution of states of finitely many particles in terms of correlation operators governed by the von Neumann hierarchy, we derive the nonlinear quantum Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy for marginal correlation operators that is adopted to the description of quantum infinite‐particle systems as well. A nonperturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed. Copyright © 2013 John Wiley & Sons, Ltd.
The rigorous evolution equation for the one-body density matrix n ( t ) describing a quantum system of interacting particles is not closed according to the investigations of Bogoliubov, Born, Green, Kirkwood, Yvon and others (B-B-G-K-Y hierarchy). The analysis of this equation is made in terms of connected diagrams. It is found that in the limits of large volume and time this equation can be reduced to a generalized Boltzmann equation which is a closed equation with respect to n ( t ) and which can rigorously describe linear and non-linear transport coefficients.
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.