J. Galvão Ramos

A nonequilibrium ensemble formalism: Criterion for truncation of description

In the framework of a nonequilibrium statistical ensemble formalism, consisting of the so-called Nonequilibrium Statistical Operator Method, we discuss the question of the choice of the space of thermohydrodynamic states. We consider in particular the relevant question of the truncation of description (reduction of the dimension of the state space). A criterion for justifying the different levels of truncation is derived. It depends on the range of wavelengths and frequencies which are the relevant ones for the characterization, in terms of normal modes, of the thermohydrodynamic motion in a nonequilibrium open system. Applications to the cases of thermal-sensitive resins and of n-doped polar semiconductors are done, numerical results are presented, and experimental observation is discussed.

Informational Statistical Thermodynamics

The term ist is applied to the theory that provides a statistical-mechanical-founded irreversible thermodynamics on the basis of Jaynes’ Predictive Statistical Mechanics with the accompanying maximum entropy formalism (MaxEnt). The use of the underlying ideas of MaxEnt to derive macroscopic properties for systems in nonequilibrium were pursued by several authors. After Jaynes’ original papers were published [106] there followed applications in which the general features of irreversibility were discussed [30, 139]. Kinetic and transport equations using MaxEnt were obtained for very specific cases [140, 141], afterwards brought together into a unifying scheme by Lewis [142] (See also references [143, 144]). Connections with cit are due to several authors, in particular to Zubarev [145]. Finally, we should mention Nettleton’s use of Maxent to obtain a description of conduction of heat in dense fluids [146, 147], and a generalized Grad-type foundation for eit with altered thermodynamic forces [40, 4 l].Nettleton pursued this line in further papers [42, 148, 149]. A version of ist founded on MaxEnt-Nesom, that is, a microscopic approach to phenomenological irreversible thermodynamics based on such particular nonequilibrium ensemble formalism is described in references [133, 134]. It provides, as a particular case, the connection of eit of the first chapter and the nesom of chapter 2 [14].

A NONEQUILIBRIUM STATISTICAL ENSEMBLE FORMALISM MaxEnt–NESOM: Basic Concepts, Construction, Application, Open Questions and Criticisms

We describe a particular approach for the construction of a nonequilibrium statistical ensemble formalism for the treatment of dissipative many-body systems. This is the so-called Nonequilibrium Statistical Operator Method, based on the seminal and fundamental ideas set forward by Boltzmann and Gibbs. The existing approaches can be unified under a unique variational principle, namely, MaxEnt, which we consider here. The main six basic steps that are at the foundations of the formalism are presented and the fundamental concepts are discussed. The associated nonlinear quantum kinetic theory and the accompanying Statistical Thermodynamics (the Informational Statistical Thermodynamics) are very briefly described. The corresponding response function theory for systems away from equilibrium allows to connected the theory with experiments, and some examples are summarized; there follows a good agreement between theory and experimental data in the cases in which the latter are presently available. We also present an overview of some conceptual questions and associated criticisms.

On the Thermodynamics of Far-From-Equilibrium Dissipative Systems

We consider the question of the concepts of entropy and temperature in arbitrary nonequilibrium conditions in the framework of the so-called Informational Statistical Thermodynamics. This is the approach to Thermodynamics based on the statistical-mechanical foundations provided by a Gibbs ensemble-like algorithm in nonequilibrium situations. The resulting nonequilibrium temperature-like variable — dubbed as quasitemperature — is shown to be a quantity measurable with appropriate “thermometric devices”. A comparison of quasitemperatures that arise in different approximated nonequilibrium statistical-thermodynamic descriptions of the dissipative system is done. The validity of these different approximations is evaluated, and (in the framework of the theory) generalized Gibbs, Clausius, and Boltzmanns relations, as well as properties of the corresponding entropy-like function (or informational entropy in Jaynes-Shannon sense), that the theory introduces, are presented. Conceptual and physical aspects of the question are also discussed, and a partial comparison of these concepts with those arising in other approaches to irreversible thermodynamics is briefly attempted.

A Thermo-Hydrodynamic Theory Based on Informational Statistical Thermodynamics

We consider the hydrodynamic description of a fluid of particles in the context of the classical approach to the Nonequilibrium Statistical Operator Mehod. It is based on the information entropy ensemble of Predictive Statistical Mechanics, and its accompanying Informational Statistical Thermodynamics. We start with a description of the macroscopic state of the system in terms of single- and two-particle reduced dynamic density functions in phase space, and the accompanying Lagrange multipliers (intensive nonequilibrium thermodynamic variables) that the method introduces. In terms of this basic set of dynamical variables we derive the equations of evolution for the mass, momentum, and energy densities, as well a the continuity equation for the informational entropy. It is shown how these equations are to be restricted in order to recover the results of classical hydrodynamics (based on linear irreversible thermodynamics), as well as a Gibbs relation defining local equilibrium. The differences between the generalized formalism and this classical limiting case are discussed.

Derivation in a nonequilibrium ensemble formalism of a far-reaching generalization of a quantum Boltzmann theory

Within the framework of the nonequilibrium statistical ensemble formalism provided by the nonequilibrium statistical operator method, we derive a quantum Boltzmann-style transport theory of a broad scope. This is done by choosing the single- and two-particle dynamical density operators as the basic informational–statistical variables. The equations of evolution for their average values over the nonequilibrium ensemble, the nonequilibrium-reduced Dirac–Landau–Bogoliubov-type density matrices, are obtained. From the resulting generalized nonlinear quantum transport theory, after resorting to perturbative-like expansions, a far-reaching generalization of Boltzmann equation for the single-particle distribution function is derived. A type of traditional Boltzmann equation follows after using stringent approximations, whose limits of validity are evaluated.

Thermo-statistics of irreversible processes: a Boltzmann-Gibbs-style ensemble formalism

The area of Physics indicated in the title is nowadays of quite relevant interest, not only from the purely scientific point of view, but specially for its applied aspects associated to the present-time point-first-technologies. A particular research trend in the theory of irreversible processes, which are evolving in time in systems arbitrarily departed from equilibrium, is here briefly described. It consists in the construction of a Gibbs-style nonequilibrium ensemble formalism. The derivation of a nonequilibrium statistical operator is described (the variational approach of Predictive Statistical Mechanics is used). The main questions involved are presented and applications are briefly mentioned.

Response Function Theory

We deal in this chapter with the question of a response function theory, which is fundamental for providing a way to relate theory and experiment. This introduces correlation functions over the given nonequilibrium ensemble, which are inevitable for the description of physical experiments [240]. Comparison with experiments provides the only way to validate a theory, with the connection of the two — theory and experiment — being indissoluble and the basis of the scientific method as it developed since the seventeenth century. As expressed by W. T. Grandy [119], in constructing a theory a first and fundamental step “consists of some kind of observation or experiment defining both the system and the problem. This is an essential procedure preceding any calculations, without which any real problem remains ill-defined. Though perhaps obvious, the point tends to be ignored in many theoretical discussions”. This point is the most pungent when dealing with not satisfactorily settled questions, as it is the thermodynamics of irreversible systems. In particular it is worth noticing the call of Ryogo Kubo [193], who 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”. As noticed, the bridge between theory and measurement is provided by response function theory.

Theory and Experiment

We make here contact with the fundamental point in the scientific method of corroborating theory through comparison with experiment [282]. It is worth mentioning S. J. Gould’s observation that “a detail, by itself, is blind; a concept without a concrete illustration is empty [...] Darwin, who had such keen understanding of fruitful procedure in science, knew in his guts that theory and observation are Siamese twins, inextricably intertwined and continually interacting” (emphasis is ours) [116]. In particular, in the present question of statistical thermodynamics we restate the call of Riogo Kubo, who expressed 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” [193].

Maxent-Nesom for Dissipative Processes in Open Systems

We consider an open many-body system out of equilibrium, which is in contact with a set of reservoirs and under the action of pumping sources. We are essentially considering the most general experiment one can think of, namely a sample (the open system of interest composed of very-many degrees of freedom) subjected to given experimental conditions, as it is diagrammatically described in Fig. 2.1