II. Quantum Density Matrix Description of Nonextensive Systems

Abstract

A description of nonextensive systems based on the Tsallis generalization of the Boltzmann-Gibbs (BG) formalism is given in terms of quantum density matrix. After developing reasons for using the density matrix description as well as generalizations of the BG-formalism in the first section, we formulate the theory of quantum entangled states and information theory in the second section. In the third section, the maximum Tsallis entropy principle with given normalized q-mean value constraints is developed in detail, leading to quantum statistical mechanics of nonextensive systems. In the fourth section, time-dependent unitary dynamics is given. Here, the Green function theory as well as linear response theory are described in detail. A brief description of nonunitary (Lindblad) dynamics is outlined in the fifth section. In the final section, open problems and possible resolution are discussed as concluding remarks. In eight Tables, the summaries of the various sections are given with a view to give intercomparison of the present developments with the familiar ones found in the literature. In Appendix, several forms of the q-Kullback-Leibler entropy are given in seeking a guide to introduce invariance principles in the theory of nonextensive Tsallis entropy.