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Acta Physica Polonica B

Time Evolution in a Two-dimensional Ultrarelativistic-like Electron Gas by Recurrence Relations Method

Érica de Mello Silva

Published Jan 1, 2015• 9 citations
Publisher

Abstract

The complexity (or simplicity) of a system can be characterized through very many paths.A quite interesting and operational one, which we follow here, consists in making use of entropic concepts.The thermodynamical and statistical-mechanical foundations of this approach, as well as its (analytical, computational, observational, and experimental) applications for natural, artificial and social systems will be briefly reviewed.

DOI
10.5506/aphyspolb.46.1135

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The complexity (or simplicity) of a system can be characterized through very many paths.A quite interesting and operational one, which we follow here, consists in making use of entropic concepts.The thermodynamical and statistical-mechanical foundations of this approach, as well as its (analytical, computational, observational, and experimental) applications for natural, artificial and social systems will be briefly reviewed.
We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numerically exact diagonalization method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations.
Bose-Einstein condensation (BEC) represents a remarkable phase transition, characterized by the formation of a single quantum subsystem. As a result, the statistical properties of the condensate are highly unique. In the case of a Bose gas, while the mean number of condensed atoms is independent of the choice of statistical ensemble, the microcanonical, canonical (CN), or grand CN (GC) variances differ significantly among these ensembles. In this paper, we review the progress made over the past 30 years in studying the statistical fluctuations of BECs. Focusing primarily on the ideal Bose gas, we emphasize the inequivalence of the Gibbs statistical ensembles and examine various approaches to this problem. These approaches include explicit analytic results for primarily one-dimensional systems, methods based on recurrence relations, asymptotic results for large numbers of particles, techniques derived from laser theory, and methods involving the construction of statistical ensembles via stochastic processes, such as the Metropolis algorithm. We also discuss the less thoroughly resolved problem of the statistical behavior of weakly interacting Bose gases. In particular, we elaborate on our stochastic approach, known as the hybrid sampling method. The experimental aspect of this field has gained renewed interest, especially following groundbreaking recent measurements of condensate fluctuations. These advancements were enabled by unprecedented control over the total number of atoms in each experimental realization. Additionally, we discuss the fluctuations in photonic condensates as an illustrative example of GC fluctuations. Finally, we briefly consider the future directions for research in the field of condensate statistics.