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International Journal of Quantum Chemistry

Generalized virial relations and the theory of subdynamics

C. H. Obcemea, Piotr Froelich, Erkki Brändas

Published Jun 19, 2009• 0 citations
Publisher

Abstract

In this article, we discuss the implication of the generalized virial relations in the spectral analysis of Liouville operators. In particular, we refer to the existence problem of the analytic continuation of these superoperators and their resolvents occurring in the reduced dynamics description of open systems. For completeness, we outline the main ideas of the subdynamics approach.

DOI
10.1002/qua.560200872

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In this article, we discuss the implication of the generalized virial relations in the spectral analysis of Liouville operators. In particular, we refer to the existence problem of the analytic continuation of these superoperators and their resolvents occurring in the reduced dynamics description of open systems. For completeness, we outline the main ideas of the subdynamics approach.
A direct approach for the generalized reduced Liouville equation of motion decoupling problem associated with open quantum systems within the superoperator formalism is presented. The procedure is based on inversion of the perturbation series for the energy representation of the density operator so as to obtain one for the proper self-energy fields that emerge as a consequence of the analytic character of the associated spectral representation. Thus, the perturbation series that arises from the iteration of the energy-dependent matrix elements hierarchy involved in the statistical operator allows, upon further expansion of the inverse of such series, to get formally exact expressions for the corrections to all orders of the self-energy fields. The lower order corrections of these fields are discussed in terms of resonant and nonresonant contributions. The present approach provides matrix equations that show the close relation between the environment effects represented by the self-energy fields and the relaxation kernel that drives the system–reservoir interaction. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 280–290, 2000
The concept of subdynamics projectors Π introduced by the Brussels group in the treatment of systems in statistical mechanics remains central in the construction of the non-unitary transformation theory or in the theory of complex spectral representations. Originally only one of such subdynamics, II, was supposed to persist asymptotically. When the evolution of the systems presents temporal behaviours with long tails, the usefulness of this subdynamics concept, and even its existence, is to be questioned. To catch the nature of the problem, we analyze simple systems with a non exponential decay. We indicate how the corresponding operators may effectively be constructed and we examine the validity of the asymptotic projection.
We study some mathematical problems posed in nonequilibrium statistical mechanics and subdynamics theory developed by Prigogine and coworkers. We study in the superspace of the Hilbert–Schmidt operators the solution of the Liouville–von Neumann equation. The application in this frame of the spectral deformation methods yields expression of analytic continuations of a class of matrix elements of the resolvent of the Liouville–von Neumann operator, and this allows the analytic continuation of the collision and destruction operators. However, it is impossible to obtain simultaneously analytic continuation of the creation operator in this frame. The above results will be used in the second part of the article in order to study the pseudo-Markovian equation.
A generalized time-dependent perturbation theory is derived for superoperators. Instead of using the ``standard'' breakup of the Hamiltonian into a known zeroth order term and a correction, we use the approximate superpropagator to define the correction superoperator which is then used to obtain a series representation of the exact Liouville operator. The theory reduces to known limits and may be used for a perturbation expansion of classical Wigner and Husimi dynamics as well as for recent phase-space-based semiclassical approximations. The theory is demonstrated for a model quartic potential.
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.