Neste artigo foi deduzida de maneira detalhada e didática a expressão da distribuição de velocidades relativística de Maxwell-Jüttner. Esta distribuição foi comparada com a bem conhecida distribuição de velocidades de Maxwell-Boltzmann, a qual não leva em consideração efeitos relativísticos. Aplicações e comparações entre as duas distribuições foram realizadas em três situações: i ) prótons à temperatura da superfície solar, ii ) elétrons à temperatura do critério de Lawson, e iii ) prótons à temperatura de quasares. Verificou-se que o valor da razão entre a massa de repouso da partícula considerada e a temperatura do sistema determina a necessidade de se considerar ou não um tratamento relativístico para a função de distribuição de velocidades do sistema.

In this paper, the expression of the relativistic Maxwell-Jüttner velocity distribution was deduced in a detailed and didactic way. This distribution was compared with the well-known Maxwell-Boltzmann distribution, which does not take into account relativistic effects. Applications and comparisons between the two distributions were carried out in three situations: i ) protons at solar surface temperature, ii ) electrons at Lawson’s criterion temperature, and iii ) protons at quasar temperature. It was found that the value of the ratio between the rest mass of the particle considered and the temperature of the system determines the need to consider or not a relativistic treatment for the velocity distribution function of the system.

Resumo Os Sistemas Dinâmicos Complexos são sistemas com propriedades emergentes inesperadas resultantes de processos sinérgicos de seus componentes. Estes sistemas aparecem na física e na química e desempenham um papel fundamental em sistemas biológicos. Tais sistemas requerem um tratamento teórico em termos de “Modelagem Matemática”, com os formalismos estatísticos sendo de grande relevância. Apresentamos aqui um artigo explanatório descrevendo e discutindo o tema. Apresentamos uma revisão do assunto com particular atenção à Teoria da Informação em uma abordagem de Shannon-Jaynes no espírito da proposta de inferência científica de Jeffreys. Enfatiza-se a difícil questão da presença das “hidden constraints” e a introdução de estatísticas não convencionais que surgem no âmbito da “Teoria da Informação”.

Abstract The “Complex Dynamical Systems” are systems with unexpected emerging properties which are the result of synergetic processes of their components. These systems appear in condensed matter physics and chemistry, as well as, playing a fundamental role, in biological systems. They require a theoretical treatment in terms of “Mathematical Modelling”, with statistical formalisms being of large relevance. We present here a feature-like article describing and discussing the question. Particular attention has been done on the basis of Information Theory in a Shannon-Jaynes approach in the spirit of Jeffreys' scientific inference proposal. It is emphasized the difficult question of presence of “hidden constraints”, and the introduction of nonstandard statistics arising in the realm of “Information Theory”.

<p>Neste artigo consideramos os aspectos gerais dos temas emergentes na segunda metade do século passado, de complexidade, auto-organização e informação. Os três estão ligados pelo fato de estarem envolvidos na “teoria de sistemas dinâmicos”, e sobre esse ponto enfocamos a descrição aqui apresentada. Descreve-se o surgimento de comportamento complexo em tais sistemas dinâmicos em geral quando governados por uma teoria cinética não linear, a emergência de estruturas dissipativas sinergéticas à la Prigogine-Haken, e a teoria de informação via o enfoque de inferência científica proposto por Jeffreys-Jaynes.</p>

<p>We describe certain general aspects of the topics complexity, self-organization and information, which emerged during the second half of the last century. They are entailed to the “theory of dynamical systems”, and it is from such point that we focus the description here presented. We describe how complex behavior arises in dynamical systems in general when they are governed by a non-linear kinetic theory as well as the emergence of self-organized structures (in the sense of the synergetic self-organized dissipative structures ”à la Prigogine-Haken”), and the theory of information through the point of view of the scientific inference of Jeffreys-Jaynes.</p>

The nowadays notable development of all the modern technology, fundamental for the progress and well being of world society, imposes a great deal of stress in the realm of basic Physics, more precisely on Thermo-Statistics. We do face situations in electronics and optoelectronics involving physical-chemical systems far-removed-from equilibrium, where ultrafast (in pico- and femto-second scale) and non-linear processes are present. Further, we need to be aware of the rapid unfolding of nano-technologies and use of low-dimensional systems (e.g., nanometric quantum wells and quantum dots in semiconductor heterostructures). All together this demands having an access to a Statistical Mechanics being efficient to deal with such requirements. It is worth noticing that the renowned Ryogo Kubo once stated that "statistical mechanics has been considered a theoretical endeavor. However, statistical mechanics exists for the sake of the real world, not for fictions. Further progress can only be hoped by close cooperation with experiment". Moreover, one needs to face the study of soft matter and fluids with complex structures (usually of the average self-affine fractal-like type). This is relevant for technological improvement in industries like, for example, that of polymers, petroleum, cosmetics, food, electronics and photonics (conducting polymers and glasses), in medical engineering, etc. It is then required to introduce a thermo-hydrodynamics going well beyond the classical (Onsagerian) one. Moreover, in the both type of situations above mentioned there often appear difficulties of description and objectivity (existence of so-called "hidden constraints"), which impair the proper application of the conventional ensemble approach used in the general, logically and physically sound, and well established Boltzmann-Gibbs statistics. A tentative to partially overcome such difficulties consists in resorting to non-conventional approaches. Here we briefly describe the construction of a Non-Equilibrium Statistical Ensemble Formalism (NESEF) that can deal, within a certain degree of success, with the situations above described. Several particular instances involving experimental observations and measurements in the area of semiconductor physics and in physics of fluids, which were analyzed in the context of the theory, are summarized. They comprise the cases of ultrafast optical spectroscopy; optical and transport processes in low-dimensional complex semiconductors; nonlinear transport in doped highly-polar semiconductors (of use in "blue diodes") under moderate to high electric fields; nonlinear higher-order thermo-hydrodynamics in fluids under driven flow, in normal solutions and in complex situations as in solutions of polymers, micelles, DNA, and in microbatteries.

A theoretical study of the mobility of n-doped III-Nitrides in wurtzite phase is reported. We have determined the nonequilibrium thermodynamic state of the bulk n-InN, n-GaN, and n-AlN systems - driven far away from equilibrium by a strong electric field - in the steady state, which follows after a very fast transient. For this we solve the set of coupled nonlinear integro-differential equations of evolution of the nonequilibrium thermodynamic variables, for the three materials, to obtain their steady state values. The dependence of the mobility (which depends on the nonequilibrium thermodynamic state of the sample) on the electric field strength and the concentration (of electrons and impurities) is derived, which decreases with the increase of the electric field strength and the concentration of carriers, evidencing the influence of the nonlinear transport involved.

We contribute here a theoretical study of the electron mobility in n-doped GaN, InN, and AlN at moderate to high electric fields. We solve the set of coupled nonlinear integro-differential equations of evolution to obtain the steady-state values of the basic intensive nonequilibrium thermodynamic variables for the three materials. The regions with ohmic and non-ohmic behavior in the electron drift velocity dependence on the electric field strength are characterized in the three nitrides. The electron mobility is calculated, and it is shown that the larger corresponds to InN, and the smaller to AlN.