(See also the book on Nonequilibrium Statistical Mechanics by R. Luzzi et al. recently published in the series FUNDAMENTAL THEORIES OF PHYSICS for a comprehensive review of the subject , which emphasizes the connection between theory and experiments)

    The objective of the statistical mechanics of far-from-equilibrium systems is to determine the thermodynamic properties and the temporal evolution of macroscopic quantities in terms of the dynamical laws that govern the movement of the particles that form the system. In this sense, the study of macroscopic states of nonequilibrium systems has to face much more serious difficulties than those present in the case of equilibrium. The need to consider the temporal evolution implies that the reversibility of the microscopic dynamics has to be reconciled with the irreversibility observed in nature. The transport equations obtained out of a microscopic theory must reflect this irreversibility and must describe, in particular, the evolution to the equilibrium in isolated natural systems. The necessity to describe both temporal and spatial evolution requires a more detailed study than in equilibrium conditions, where most of the problems can be described by an appropriate state function. However, in most of the irreversible problems, there is no control over the system and, except in stationary states, it is not possible to describe the phenomenon with a non-equilibrium thermodynamic potential.

    It can be said that non-equilibrium statistical thermodynamics had its origin during the last century with the works by Boltzmann who developed the basis for a kinetic theory of gases valid for diluted systems. These works allowed, among other things, the development of a transport theory, the most important result of which is the well-known Boltzmann equation. The H-theorem was other of the first and very important results in the field, showing for the first time how a kinetic theory allows it to describe evolution towards equilibrium. The extension of this theory to non-diluted systems was developed by several authors and followed essentially two directions: i) a generalization of the kinetic theory of gases, what led to the equations of the BBGKY hierarchy (of which the Boltzmann equation is the lowest order approximation) and ii) a generalization of the Brownian motion in which the cumbersome Langevin-like equations arising out of the microscopic mechanical laws are treated using appropriate statistical approximations (this leads to the Mori's correlation function formalism and the master equation method).

    A different approach for the development of a statistical mechanics suited for systems arbitrarily away from equilibrium is based on Gibbs' ensemble formalism. How its application can be justified and what its physical meaning is, are questions with no satisfactory answers even under equilibrium conditions, although its success motivates extending its application to systems arbitrarily away from equilibrium. The extension to these systems has associated a fundamental question: what is the appropriate ensemble for a given non-equilibrium situation?

    The Nonequilibrium Statistical Operator Method (NESOM) is a technique based on Gibbs ensemble formalism [1]. Several approaches for this method are available and it has been showed that they can be unified in a unique variational principle [2]. This principle (that leads to the construction of the nonequilibrium ensemble) consists in the maximization of a particular function, defined in the system, at each instant of time. This function is named Informational Entropy (or Gibbs statistical entropy) and its maximization must be carried out taking into account the restrictions imposed by the relevant information about the system. Therefore,  the NESOM based on this technique of maximization of information entropy will be called MaxEnt-NESOM. The method offers a criterion for choosing the suited ensemble for a given nonequilibrium situation, answering, in this way, the fundamental question stated in the previous paragraph. The correctness of this choice in a particular problem (and, implicitly, the validity of the variational principle) can only be verified a posteriori by comparing the theoretical results with experimental results. MaxEnt-NESOM allows it to obtain transport equations that are non-linear and non-local both in space and time, i.e., it has incorporated memory effects and spatial correlations.

    The same way as equilibrium statistical mechanics offers microscopic foundations to equilibrium thermodynamics (or thermostatics), statistical mechanics of irreversible systems must form the basis of a non-equilibrium thermodynamics. However, the traditional study of irreversible thermodynamics has followed paths independent on the microscopic approach, and has been based mainly on phenomenological hypothesis. Both non-equilibrium thermodynamics and statistical mechanics theories are currently in development and involve serious conceptual difficulties. Among the phenomenological approaches for nonequilibrium thermodynamics we can mention: i) Classical Irreversible Thermodynamics (CIT); ii) Rational Thermodynamics (RT) and iii) Extended Irreversible Thermodynamics (EIT). CIT [3] is based on the hypothesis of local equilibrium, assuming as valid many of the concepts and relations of equilibrium thermodynamics, i.e., thermostatics is valid in any small elements of volume DV(r) at position r and at any instant of time t. To close the theory the so-called constitutive relations are included that relate the flows with the basic thermodynamic variables (as, e.g., Fick's and Fourier's laws). The most serious limitations and problems of CIT are related with the local equilibrium hypothesis and with the mathematical character of the constitutive relations that describe perturbations propagating at infinite velocity (like the diffusion equations for mass and heat, that are described by differential equations of the parabolic type).

    Both RT and EIT were developed to deal with systems arbitrarily away from equilibrium, therefore extending the scope of CIT. Rational Thermodynamics [4] abandons the concept of local equilibrium and deals with response functionals that describe the system, at time t, by the values of the thermodynamic variables determined along all the previous evolution of the system (at times t'<t), introducing this way the concept of memory.

    The Extended Irreversible Thermodynamics [5] is the most complete and accepted formulation, and allows it to deal in a satisfactory way with many problems. EIT generalizes CIT in several aspects: it goes beyond the hypothesis of local equilibrium, it attempts to obtain a better agreement with the experiment and a better relationship with nonequilibrium statistical mechanics. Moreover, by means of a more rigorous mathematical formalism, eliminates the conceptual difficulties related to infinite velocity of propagation that CIT introduces (the equations describing mass and heat propagation are now of the hyperbolic type, producing wave-like behavior).

    The statistical operator method has led to the development of a thermodynamic theory with microscopic basis. The thermodynamics associated to MaxEnt-NESOM is called Informational Statistical Thermodynamics [6] and it has been shown to have a close relation with the phenomenological thermodynamic theories described above [7].

    Here we study both the statistical mechanics and the irreversible thermodynamics of a many-body system taken arbitrarily away from equilibrium. After general considerations and particular studies on the statistical thermodynamics we perform an extensive application to physical systems of both theoretical and technological relevance. In particular, we study a direct-gap, polar semiconductor in contact with a thermal bath, and kept far from equilibrium by a continuous excitation with an external source of electromagnetic radiation of broad spectrum (the particular case of excitation by laser is obtained as a limit case in which the spectrum of frequencies is modeled by a delta-like function centered at the energy of the laser photons). In the situation described above electrons in the valence band are excited and moved to the conduction band by absorption of a photon of the external radiation. In the electron-hole representation we have two fluids of quantum quasi-particles that dissipate their energy by means of two relaxation mechanisms: i) electron-hole recombination, with the emission of electromagnetic energy, and ii) energy transfer to the crystal lattice by electron-phonon interaction. In this excitation/relaxation game the system is taken and kept out of equilibrium and can, eventually, attain a stationary state after transients have occurred. The focused here is in the stationary states rather than in the temporal evolution (with the exception of the dynamical study carried out in Chapter 3, where we study the asymptotic temporal evolution of the system near the final stationary states). It is in this situation where the phenomena we are interested here occur (and can be observed), i.e., the effects on the optical properties of the system of the non-equilibrium conditions and their concomitant dissipative effects, and the possibility of emergence of complex behavior consistent with the formation of spatial structures in semiconductors under intense electromagnetic radiation. In the situation described above, we have different interacting subsystems composed of many quasi-particles: electrons, holes, photons (from the external radiation and from the product of the internal recombination), optical phonons (both longitudinal and transversal) and acoustic phonons. Therefore, the system described has the ideal component for carrying out a detailed analysis at the microscopic and thermodynamic levels of the dissipative (irreversible) processes in a physical system of experimental and theoretical relevance.

    The physics of semiconductors appears as an ideal area to test ideas and methods of nonequilibrium statistical thermodynamics because of the possibility of performing experimental studies in conditions of strong non-quilibrium and the availability of many advanced and accurate measurement techniques. The microscopic formalism of NESOM has been applied extensively to the area of semiconductors and good agreement with experimental results was obtained [8-14]. Other areas of application of the theory is in biosystems, in particular, in quasi-one-dimensional systems like proteins, biopolymers, etc. [15].

[1] D N Zubarev, V Morosov and G Ropke, Statistical Mechanics of Non-Equilibrium Processes, Vols 1 and 2 (Akademie Verlag, Berlin, 1996 and 1997, respectively)

[2] R Luzzi and A R Vasconcellos, On the Non-Equilibrium Statistical Operator Method, Fortschr. Phys./Prog. Phys. 38, 887 (1990)

[3] I Prigogine, Etude Thermodinamique des Phenomenes Irreversibles (Desoer, Liege, 1947) [Introduction to the Thermodynamics of Irreversible Processes (Thomas, New York, 1955; Wiley-Interscience, New York, 1961)]; S de Groot and P Mazur, Nonequilibrium Thermodynamics (North-Holland, Amsterdam, 1962)

[4] C Truesdell, Rational Thermodynamics (McGraw-Hill, Nww York, 1969; 2nd Ed, Springer, New York, 1984)

[5] D Jou, J Casas-Vazquez and G Lebon, Extended Irreversible Thermodynamics (Springer, Berlin, 1993; 2nd Ed, 1996)

[6] A Hobson, Irreversibility and Information in Mechanical Systems, J Chem Phys 45, 1352 (1966); E T Jaynes, Information Theory and Statistical Mechanics I, Phys Rev 106, 620 (1957); II, Phys Rev 108, 171 (1957); L S Garcia-Colin, A R Vasconcellos and R Luzzi, On Informational Statistical Thermodynamics, J Non-Equilib Thermodyn 19, 24 (1994)

[7] M A Tenan, A R Vasconcellos and R Luzzi, Mechano-Statistical Foundations for Generalized Nonequilibrium Thermodynamics, Fortschr Phys/Prog Phys 45, 1 (1997)

[8] D K Ferry, H L Grubin and G J Iafrate, Transient Transport in Semiconductors and Submicron Devices, in Semiconductor Probed by Ultrafast Laser Spectroscopy, Vol I, Ed. R R Alfano (Academic, New York, 1984)

[9] X L Lei, D Y Xing, M Liu, C S Ting and J L Birman, Nonlinear Electronic Transport in Semiconductor System with Two Types of Carriers: Application to GaAs, Phys Rev B 36, 9134 (1987); D Y Xing, P Hu and C S Ting, Balance Equations for Steady-State Hot-Electron Transport in the Approach of the Nonequilibrium Statistical Operator, Phys Rev B 35, 6379 (1987); L Liu, D Y Xing, C S Ting and W T Xu, Hot-Electron Transport for Many-Valley Semiconductors by the Method of Nonequilibrium Statistical Operator, Phys Rev B 37, 2997 (1988); D Y Xing and C S Ting, Green's Function Approach to Transient Hot-Electron Transport in Semiconductors under Uniform Electric Field, Phys Rev B 35, 3971 (1987)

[10] V N Freire, A R Vasconcellos and R Luzzi, Nonlinear Ultrafast Transient Transport in Polar Semiconductors, Phys Rev 39, 13264 (1988)

[11] R Luzzi and L C Miranda, Optical Properties of Semiconductors under Intense Laser Radiation, in Physics Reports Reprint Book Series, Vol 3 (North-Holland, Amsterdam, 1978) pp. 423-453; R Luzzi and A R Vasconcellos, Ultrafast Transient Response of Nonequilibrium Plasma in Semiconductors, in Semiconductors Processes Probed by Ultrafast Laser Spectroscopy, Vol I, Ed R R Alfano (Academic, New York, 1984); R Luzzi, Ultrafast Relaxation Processes in Semiconductors, in High Excitation and Short Pulse Phenomena, Ed M H Pilkhun (North-Holland, Amsterdam, 1985)

[12] A C Algarte, A R Vasconcellos and R Luzzi, Ultrafast Phenomena in Semiconductors, Braz J Phys 26, 543 (1996)

[13] A C Algarte, A R Vasconcellos and R Luzzi, Kinetics of Hot Elementary Excitations in Photoinjected Polar Semiconductors, Phys Stat Sol B 173, 487 (1992)

[14] A C Algarte, A R Vasconcellos and R Luzzi, Diffusion of Photoinjected Carriers in Nonequilibrium Polar Semiconductors, Solid State Commun 87, 299 (1993); A R Vasconcellos, A C Algarte and R Luzzi, Diffusion of Photoinjected Carriers in Plasma in Nonequilibrium Semiconductors, Phys Rev B 48, 10873 (1993)

[15] M V Mesquita, A R Vasconcellos and R Luzzi, Phys Rev E 48, 4049 (1993)