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Brazilian Journal of Physics

Field-theoretical methods and nonextensive systems

R. S. Mendes

Published Mar 1, 1999• 4 citations
Publisher

Abstract

Traditional field-theoretical methods to study extensive many-particle systems are generalized to discuss nonextensive situations. In particular, generalizations of Green functions, path integral, and Gaussian integration are performed in the context of nonextensive Tsallis statistical mechanics. These developments employ integral representations that connect the usual and the generalized cases.

DOI
10.1590/s0103-97331999000100007

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Traditional field-theoretical methods to study extensive many-particle systems are generalized to discuss nonextensive situations. In particular, generalizations of Green functions, path integral, and Gaussian integration are performed in the context of nonextensive Tsallis statistical mechanics. These developments employ integral representations that connect the usual and the generalized cases.
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This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
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.
In the first three chapters we have considered problems related to the physical behaviour of one or a few particles (or some other physical objects which can be effectively described by one or a few (quasi)particles as, e.g., the random walk model in polymer physics considered in chapter 1). Though quantum field theory describes systems with an arbitrary number of degrees of freedom (an arbitrary number of (quasi)particles), in chapter 3 we actually applied it to systems with a restricted and small number of particles (e.g., for the description of the scattering process for a few particles). Another way to express this fact is to say that in chapter 3 we have considered field theories at zero temperature. However, the majority of realistic systems contain many identical (indistinguishable) particles such as atoms, electrons, photons etc. An attempt to describe these systems in terms of the individual trajectories of all particles is absolutely hopeless. Instead, we are interested in the collective behaviour of systems and describe them in terms of partition functions, mean values, correlation functions, etc. The methods of derivation, analysis and calculation of such collective characteristics constitute the subject of classical and quantum statistical mechanics.
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.