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Physical Review Letters

Quantum Statistical Mechanics for Nonextensive Systems: Prediction for Possible Experimental Tests

A. K. Rajagopal, R. S. Mendes, Ervin Kaminski Lenzi

Published May 4, 1998• 70 citations
Publisher

Abstract

The traditional basis of description of the many-particle systems in terms of the Green functions is here generalized when the system is nonextensive by incorporating the Tsallis form of the density matrix, indexed by a nonextensive parameter, $q$. This extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the $q$ averages.

DOI
10.1103/physrevlett.80.3907

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The traditional basis of description of the many-particle systems in terms of the Green functions is here generalized when the system is nonextensive by incorporating the Tsallis form of the density matrix, indexed by a nonextensive parameter, qqq. This extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the qqq averages.
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The traditional basis of description of many-particle systems in terms of Green functions is here generalized to the case when the system is nonextensive, by incorporating the Tsallis form of the density matrix indexed by a nonextensive parameter q. This is accomplished by expressing the many-particle q Green function in terms of a parametric contour integral over a kernel multiplied by the usual grand canonical Green function which now depends on this parameter. We study one- and two-particle Green functions in detail. From the one-particle Green function, we deduce some experimentally observable quantities such as the one-particle momentum distribution function and the one-particle energy distribution function. Special forms of the two-particle Green functions are related to physical dynamical structure factors, some of which are studied here. We deduce different forms of sum rules in the q formalism. A diagrammatic representation of the q Green functions similar to the traditional ones follows because the equations of motion for both of these are formally similar. Approximation schemes for one-particle q Green functions such as Hartree and Hartree-Fock schemes are given as examples. This extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the q averages.
Abstract A dynamic theory of the linear reaction from nonextensive quasi-equilibrium multibody systems to an external time-dependent perturbation is developed in quantum statistical mechanics based on the Tsallis parametric nonadditive entropy associated with the density matrix. For nonextensive quantum systems, a modification of the Kubo theory developed in quantum mechanics is proposed. The linear reaction theory is constructed based on a generalized canonical form of the density matrix obtained by maximizing the Tsallis quantum entropy by averaging the observed values over the escort distribution. Generalized expressions for admittance and response functions are presented that describe the linear response of the system to a weak external mechanical impact. The paper discusses the symmetry property for the relaxation function under time reversal and the Onsager reciprocity relation for generalized susceptibility. It is shown that these properties known in classical quantum statistics remain valid for anomalous systems.