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Physica A Statistical Mechanics and its Applications

Quantum BBGKY-hierarchies and Wigner's equation in postgalilean approximation

Yu. N. Orlov, Igor Pavlovich Pavlotsky

Published Jul 1, 1989• 9 citations
Publisher
DOI
10.1016/0378-4371(89)90380-4

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Quantum Kinetic Equations and Evolution of Many-Particle Systems

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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.

arXiv:0908.2797•Aug 19, 2009•0 citations
We present a Hamiltonian method of constructing BBGKY-like hierarchies for quantum field theories. With suitable choices, our method creates a hierarchical system of evolution equations for the k-th order reduced density matrices. These equations can be closed at finite order using methods developed for the classical BBGKY hierarchy to give non-perturbative approximations for the full quantum equations of motion. Classical observables can then be numerically computed from these approximate equations, providing an analytically tractable method of modeling high-energy environments where quantum effects play a pronounced role.
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.