On The Statistical Foundations Of Irreversible Thermodynamics

We briefly, and partially, consider aspects of the present status of phenomenological irreversible thermodynamics and nonequilibrium statistical mechanics. After short comments on Classical Irreversible Thermodynamics, its conceptual and practical shortcomings are pointed out, as well as the efforts undertaken to go beyond its limits, consisting of particular approaches to a more general theory of Irreversible Thermodynamics. In particular, a search for statistical-mechanical foundations of Irreversible Thermodynamics, namely, the construction of a statistical thermodynamics, are based on the Non-equilibrium Statistical Operator Method. This important theory for the treatment of phenomena at the macroscopic level, is based on a microscopic molecular description in the context of a nonequilibrium ensemble formalism. We draw attention to the fact that this method may be considered to be emcompassed within Jaynes' Predictive Statistical Mechanics and based on the principle of maximization of informational entropy. Finally, we describe how, in fact, the statistical method provides foundations to phenomenological irreversible thermodynamics, thus giving rise to what can be referred to as Informational Statistical Thermodynamics.474401432Brown, S.C., The caloric theory of heat (1950) Am. J. Phys., 18, p. 367Brush, S.G., (1976) The Kind of Motion We Call Heat, , North Holland, AmsterdamMeixner, J., (1968) Thermodynamic of Irreversible Processes has Many Faces, in Irreversible Aspects of Continuum Mechanics, , H. Parkus and L. I. Sedov, Eds. (Springer, Wien)see also Ref. [19] belowAnderson, P.W., More is different (1972) Science, 117, p. 393Anderson, P.W., Is complexity physics? Is it science? What is it? (1991) Phys. Today, 44 (7), p. 9Lemonick, M.D., Life, the universe, and everything (1993) Time Magazine, pp. 40-41. , February 22Nicolis, G., Physics of far-from-equilibrium systems and selforganization (1989) The New Physics, , P. Davies, Ed. (Cambridge Univ. Press, Cambridge)Ebeling, W., Muschik, W., (1993) Statistical Physics and Thermodynamics of Non-linear Nonequilibrium Systems, , World Scientific, SingaporeJaynes, E.T., Foundations of probability theory and statistical mechanics (1967) Delaware Seminar in the Foundations of Physics, , M. Bunge, Ed. (Springer, New York)Jaynes, E.T., Bayesian methods: General background (1986) Maximum Entropy and Bayesian Methods, , J. H. Justice, Ed. (Cambridge Univ. Press, Cambridge)Jaynes, E.T., On the rationale of maximum entropy methods (1982) Proc. IEEE, 70, p. 939García-Colín, L.S., Vasconcellos, A.R., Luzzi, R., On informational statistical thermodynamics (1994) J. Non-Equilib. Thermodyn., 19, p. 24Lebowitz, J.L., Boltzmann's entropy and time's arrow (1993) Phys. Today, 46 (9), p. 32Barnum, H., Driebe, D.J., Hoover, W.G., Lebowitz, J.L., (1994) Letters Section of Phys. Today, 47 (11), pp. 11-15. , NovemberPrigogine, I., Étude thermodynamique des phénomènes irreversibles (1947) Introduction to Thermodynamics of Irreversible Processes, , Desoer, Liège, Thomas, New YorkDe Groot, S.R., Mazur, P., (1962) Nonequilibrium Thermodynamics, , North Holland, AmsterdamOnsager, Reciprocal relations in irreversible processes I (1931) Phys. Rev., 37, p. 405(1931) Phys. Rev., 38, p. 2235Glansdorff, P., Prigogine, I., (1971) Thermodynamic Theory of Structure, Stability and Fluctuations, , Wiley-Interscience, New YorkTisza, L., (1991) Thermodynamics: History and Philosophy, pp. 515-522. , K. Martinas, L. Ropolyi, and P. Szegedi, Eds. (World Scientific, Singapore)Drago, A., Thermodynamics: History and Philosophy, pp. 329-345Matinas, K., Thermodynamics: History and Philosophy, pp. 285-303Eu, B.C., Kinetic theory of irreversible thermodynamics (1992) Phys. Rev. E, 51, p. 768. , Wiley, New YorkTruesdell, C., (1985) Rational Thermodynamics, , McGraw-Hill, New York [2nd enlarged edition (Springer, Berlin)]Jou, D., Casas-Vazquez, J., Lebon, G., (1993) Extended Irreversible Thermodynamics, , Springer, Berlin, first and second enlarged editions respectivelyPrigogine, I., Le domaine de validité de la thermodynamique des phenomènes irreversibles (1949) Physica, 15, p. 272Meixner, J., (1970) Critical Review of Thermodynamics, , E. B. Stuart, B. Gal-Or and A. J. Bainard, Eds. (Mono-Book, Baltimore)Müller, I., Zum paradoxon der wärmeleitungstheorie (1967) Z. Phys., 198, p. 329Rivlin, R.S., Forty years of nonlinear continuum mechanics (1984) Proc. IX Int. Congress on Rheology, , A. Garcia-Rejón and C. Rangel-Lafaile, Eds. (Universidad Autónoma Metropolitana, México)Maxwell, J.C., On the dynamical theory of gases (1867) Phil. Trans. Roy. Soc., 157, p. 49. , LondonNiven, W.R., (1965) The Collected Papers of J. C. Maxwell, , Dover, New YorkCattaneo, C., Sur une form de l'equation de la chaleur eliminant de le paradoxe d'une propagation instantaine (1958) Comp. Rend. Acad. Sci. (Paris), 247, p. 431Vernotte, P., Les paradoxes de la teorie de l'equation de la chaleur (1958) Comp. Rend. Acad. Sci. (Paris), 246, p. 3154Nettleton, R.E., Relaxation thermal theory of conduction in liquids (1960) Phys. Fluids, 3, p. 216Nettleton, R.E., Early applications of extended irreversible thermodynamics (1983) Recent Developments in Nonequilibrium Thermodynamics, , D. Jou, J. Casas-Vázquez and G. Lebon, Eds. (Springer, Berlin)Nettleton, R.E., Nonlinear reciprocity and the maximum entropy formalism (1989) Physica A, 158, p. 672Nettleton, R.E., Generalized grad-type foundation for nonlinear extended thermodynamics (1990) Phys. Rev. A, 42, p. 4622Nettleton, R.E., Statistical comparison of local and nonlocal extended thermodynamics (1991) S. Afr. J. Phys., 14, p. 27Müller, I., (1985) Thermodynamics, , Pitman, LondonMüller, I., Ruggieri, T., (1993) Extended Thermodynamics, , Springer, New YorkMüller, I., Extended thermodynamics: Past, present, future (1983) Recent Developments in Nonequilibrium Thermodynamics, , D. Jou, J. Casas-Vázquez and G. Lebon, Eds. (Springer, Berlin)Gyarmati, I., On the wave approach to thermodynamics and some problems of nonlinear theories (1977) J. Non-Equil. Thermodyn., 2, p. 233Lambermont, J., Lebon, G., On a generalization of Gibbs equation for heat conduction (1973) Phys. Rev. Lett. A, 42, p. 499Jou, D., Casas-Vázquez, J., Lebon, G., A dynamical interpretation of second order constitutive equations of hydrodynamics (1979) J. Non-Equil. Thermodyn., 4, p. 349Jou, D., Casas-Vázquez, J., Fluctuations of dissipative fluxes and the onsager-machlup function (1980) J. Non-Equil. Thermodyn., 5, p. 91Casas-Vázquez, J., Jou, D., An approach to extended irreversible thermodynamics: Fluctuation theory (1983) Recent Developments in Nonequilibrium Thermodynamics, , D. Jou, J. Casas-Vázquez and G. Lebon, Eds. (Springer, Berlin)Lebon, G., An approach to extended irreversible thermodynamics (1983) Recent Developments in Nonequilibrium Thermodynamics, , D. Jou, J. Casas-Vázquez and G. Lebon, Eds. (Springer, Berlin)Lebon, G., From classical irreversible thermodynamics to extended irreversible thermodynamics (1989) Acta Physica Hungarica, 66, p. 241. , Festschrift in honor of I. GyarmatiGarcía-Colín, L.S., (1982) Proc of the First Mexican School in Statistical Physics, , R. Peralta-Fabi, Ed. (Sociedad Mexicana de Física, MéxicoGarcía-Colín, L.S., Rodriguez, R.F., Lopez De Haro, M., Jou, D., Casas-Vázquez, J., On the foundations of extended irreversible thermodynamics (1984) J. Stat. Phys., 37, p. 465García-Colín, L.S., Extended irreversible thermodynamics and chemical kinetics (1983) Recent Developments in Nonequilibrium Thermodynamics, , D. Jou, J. Casas-Vázquez and G. Lebon, Eds. (Springer, Berlin)García-Colín, L.S., Some views on extended irreversible thermodynamics (1989) Acta Physica Hungarica, 66, p. 79. , Festschrift in honor of I. GyarmatiJou, D., Casas-Vázquez, J., Lebon, G., Extended irreversible thermodynamics (1988) Rep. Prog. Phys., 51, p. 1105García-Colín, L.S., Extended non-equilibrium thermodynamics, scope and limitations (1988) Revista Mexicana de Física, 34, p. 344García-Colín, L.S., Uribe, F.J., Extended irreversible thermodynamics beyond the linear regime: A critical overview (1991) J. Non-Equilib. Thermodyn., 16, p. 89Lebon, G., Jou, D., Casas-Vázquez, J., Questions and answers about a thermodynamic theory of the third type (1992) Contemp. Phys., 33, p. 41Jou, D., Casas-Vázquez, J., Lebon, G., Extended irreversible thermodynamics: An overview of recent bibliography (1992) J. Non-Equil. Thermodyn., 7, p. 383Casas-Vázquez, J., Jou, D., Lebon, G., (1983) Recent Developments in Nonequilibrium Thermodynamics, , Springer, BerlinBoltzmann, L., (1964) Lecture on Gas Theory, , Univ. California Press, Berkeley, CaliforniaZiman, J.M., (1960) Electrons and Phonons: The Theory of Transport Phenomena in Solids, , Clarendon, OxfordRamos, J.G., Vasconcellos, A.R., Luzzi, R., A classical approach in predictive statistical mechanics: A generalized Boltzmann-like formalism (1995) Fortschr. Phys./Prog. Phys., 43, p. 265Ford, G.W., Uhlenbeck, G.E., (1963) Lectures in Statistical Mechanics, , M. Kac, Ed. (Am. Math. Soc., Providence, Rhode Island)Jaynes, E.T., Gibbs versus Boltzmann entropies (1965) Am. J. Phys., 33, p. 391Grad, H.E., The principles of the kinetic theory of gases (1958) Handbuch der Physik III, 12. , S. Flügge, Ed. (Springer, Berlin)García-Colín, L.S., Fuentes-Martínez, G., A kinetic derivation of extended irreversible thermodynamics (1982) J. Stat. Phys., 29, p. 387Mori, H., Oppenheim, I., Ross, J., Some topics in quantum statistics (1962) Studies in Statistical Mechanics I, pp. 217-298. , J. de Boer and G. E. Uhlenbeck, Eds., (North Holland, AmsterdamJaynes, E.T., (1983) E. T. Jaynes Papers on Probability, Statistics, and Statistical Physics, , R. D. Rosenkrantz, Ed. (Dortrecht, Reidel)Robertson, B., Applications of maximum entropy to nonequilibrium statistical mechanics (1978) The Maximum Entropy Formalism, pp. 289-320. , M. Tribus and R. D. Levine, Eds., MIT Press, Cambridge, MassachusettsGrandy, W.T., Principle of maximum entropy and irreversible processes (1980) Phys. Rep., 62, p. 175Kubo, R., (1959) Lectures in Theoretical Physics, , W. Brittin, Ed. (Wiley, New York)Kubo, R., The fluctuation-dissipation theorem (1969) Many-Body Problems, , S. F. Edwards, Ed. (Benjamin, New York)Mori, H., Relaxation phenomena near the critical points (1966) Many-Body Theory, , R. Kubo, Ed. (Syokabo, Tokyo)Mori, H., Transport, collective motion and Brownian motion (1965) Prog. Theor. Phys. (Japan), 33, p. 423Mori, H., A continued-fraction representation of the time-correlation functions (1965) Prog. Theor. Phys. (Japan), 34, p. 399Zwanzig, R., (1965) Annual Review of Physical Chemistry, 16, pp. 67-102. , Academic Press, New YorkZwanzig, R., Statistical mechanics of irreversibility (1961) Lectures in Theoretical Physics, 3. , W. E. Brittin, B. W. Downs, and J. Downs, Eds. (Wiley-Interscience, New York)Zwanzig, R., Memory effects in irreversible thermodynamics (1961) Phys. Rev., 124, p. 983García-Colín, L.S., Del Rio, J.L., A unified approach for deriving kinetic equations in nonequilibrium statistical mechanics I (1977) J. Stat. Phys., 16, p. 235A unified approach for deriving kinetic equations in nonequilibrium statistical mechanics II: Approximate results (1978) J. Stat. Phys., 19, p. 109Grabert, H., (1981) Projection Operator Techniques in Nonequilibrium Statistics, , Springer, BerlinLuzzi, R., Vasconcellos, A.R., Complex behavior in condensed matter: Morphological ordering in dissipative carrier system (1997) Complexity, 2, p. 42Prigogine, I., Structure, dissipation and life (1969) From Theoretical Physics to Biology, , M. Marois, Ed. (North Holland, Amsterdam)Prigogine, I., Nicolis, G., Babloyantz, A., Thermodynamics of evolution I (1972) Phys. Today, 25 (11), p. 23Prigogine, I., Nicolis, G., Babloyantz, A., Thermodynamics of evolution II (1972) Phys. Today, 25 (12), p. 38Nicolis, I., Prigogine, I., (1989) Exploring Complexity, , Freeman, New YorkPenrose, O., Foundations of statistical mechanics (1979) Rep. Prog. Phys., 42, p. 1938Kubo, R., Oppening address to Oji seminar on nonlinear nonequilibrium statistical mechanics (1978) Prog. Theor. Phys. (Japan), (SUPPL. 64), p. 1Zwanzig, R., Where do we go from here? (1981) Perspectives in Statistical Physics, , H. J. Raveché, Ed., pp. 123-124 (North Holland, Amsterdam)Dorfmann, J.R., Advances and challenges in the kinetic theory of gases (1981) Physica A, 106, p. 77García-Colín, L.S., Green, M.S., The chapman-enskog solution of the generalized Boltzmann equation (1966) Physica, 32, p. 450Bogoliubov, N.N., (1967) Lectures in Quantum Statistics I, , Gordon and Breach, New YorkKreuzer, H.J., (1981) Nonequilibrium Thermodynamics and Its Statistical Foundations, , Clarendon, OxfordMori, H., Statistical-mechanical theory of transport in fluids (1958) Phys. Rev., 112, p. 1829Correlation function method for transport phenomena (1959) Phys. Rev., 115, p. 298Forster, D., (1975) Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, , Benjamin, Readings, MALebowitz, L.L., Montroll, E.W., Preface in studies (1983) Statistical Mechanics X. Nonequilibrium Phenomena I: The Boltzmann Equation, , North Holland, AmsterdamMcLennan, J.A., Nonlinear effects in transport theory (1961) Phys. Fluids, 4, p. 1319Peletminskii, S.V., Yatsenko, A.A., Contribution to the quantum theory of kinetic and relaxation processes (1967) Zh. Eksp. Teor. Fiz., 53, p. 1327. , [Soviet Phys. - JETP 26, 773 (1968)Zubarev, D.N., Kalashnikov, V.P., External properties of the nonequilibrium statistical operator (1969) Teor. Mat. Fiz., 1, p. 137. , [Theor. Math. Phys. (USSR) 1, 108 (1970)]Zubarev, D.N., (1971) Neravnovesnaia Statisticheskaia Termodinamika, , Izd. Nauka, Moscow, [English transl.: Nonequilibrium Statistical Thermodynamics (Plenum - Consultants Bureau, New York, 1974)]Zubarev, D.N., Morosov, V., Röpke, G., Statistical mechanics of nonequilibrium processes (1996) Basic Concepts, Kinetic Theory, 1. , Akademie Verlag, Berlin(1997) Relaxation and Hydrodynamic Processes, 2. , Akademie Verlag, BerlinLuzzi, R., Vasconcellos, A.R., On the nonequilibrium statistical operator method (1990) Fortschr. Phys./Prog. Phys., 38, p. 887Zubarev, D.N., The method of the non-equilibrium statistical operator and its applications (1970) Fortschr. Phys./Prog. Phys., 18, p. 125Zubarev, D.N., Novikov, M.Yu., Die boltzmanngleichung und mögliche wege zur entwicklung dynamischer methoden in der kinetischen theorie (1973) Fortschr. Phys./Prog. Phys., 21, p. 703Zubarev, D.N., Modern methods of the statistical theory of nonequilibrium processes (1980) Reviews of Science and Technology: Modern Problems of Mathematics, 15, pp. 131-226. , in RussianR. B. Gramkreludze, Ed. (Izd. Nauka, Moscow) [English transl.: J. Soviet Math. 16, 1509 (1981)]Zubarev, D.N., (1993) Orbituary in Teor. Mat. Fiz., 96, p. 321. , [Theor. Math. Phys. 96, 995 (1994)]Jaynes, E.T., Macroscopic prediction (1985) Complex Systems: Operational Approaches, , H. Haken, Ed. (Springer, Berlin)Jaynes, E.T., Predictive statistical physics (1986) Frontiers of Nonequilibrium Statistical Physics, , G. T. Moore and M. O. Scully, Eds. (Plenum, New York)Jaynes, E.T., Clearing up misteries: The original goals (1989) Maximum Entropy and Bayesian Methods, , J. Stalling, Ed. (Kluwer, Dordrecht)Jaynes, E.T., Notes on present status and future prospects (1990) Maximum Entropy and Bayesian Methods, , W. T. Grandy and L. H. Schick, Eds. (Kluwer, Dordrecht)Jaynes, E.T., Information theory and statistical mechanics I (1957) Phys. Rev., 106, p. 620Information theory and statistical mechanics II (1957) Phys. Rev., 108, p. 171Shannon, G.E., Weaver, W., (1948) The Mathematical Theory of Communication, , Univ. Illinois Press, UrbanaLuzzi, R., Vasconcellos, A.R., Ramos, J.G., Statistical foundations of irreversible thermodynamics Texte zur Physik Series, , W. Ebeling, Ed. (Teubner, Berlin, in press)Ramos, J.G., Vasconcellos, A.R., García-Colín, L.S., A thermo-hydrodynamic theory based on informational statistical thermodynamics (1997) Braz. J. Phys., 27, p. 585Grandy, W.T., (1987) Foundations of Statistical Mechanics I: Equilibrium Systems, , Reidel, DordrechtGrandy, W.T., (1988) Foundations of Statistical Mechanics II: Nonequilibrium Phenomena, , Reidel, DordrechtAkhiezer, A.I., Peletminskii, S.V., (1981) Methods of Statistical Physics, , Pergamon, OxfordMcLennan, J.A., (1989) Introduction to Nonequilibrium Statistical Mechanics, , Prentice-Hall, Englewoods CliffsKirkwood, J.C., The statistical mechanical theory of transport processes (1946) J. Chem. Phys., 14, p. 180(1947) J. Chem. Phys., 15, p. 72Green, M.S., Markoff random processes and the statistical mechanics of time-dependent phenomena I (1952) J. Chem. Phys., 20, p. 1281Markoff random processes and the statistical mechanics of time-dependent phenomena II (1954) J. Chem. Phys., 22, p. 398Robertson, R., Equations of motion in nonequilibrium statistical mechanics (1966) Phys. Rev., 144, p. 151(1968) Phys. Rev., 160, p. 175(1968) Phys. Rev., 166, p. 206Kalashnikov, V.P., Linear relaxation equations in the nonequilibrium statistical operator method (1978) Theor. Math. Phys., 34, p. 263Vstovskii, V.P., Projection formalism in the theory of irreversible process (1975) Theor. Math. Phys., 21, p. 1214Sergeev, M.V., Generalized transport equations in the theory of irreversible process (1975) Theor. Math. Phys., 21, p. 1234Lauck, E., Vasconcellos, A.R., Luzzi, R., A nonlinear quantum transport theory (1990) Physica A, 168, p. 789Hawkings, S., (1989) 1990 -Yearbook of Science and Future, , Encyclopaedia Britannica, ChicagoLestiene, R., Entropy, mechanical time and cosmology arrow (1980) Scientia, 113, p. 313Prigogine, I., (1980) From Being to Becoming, , Freeman, San FranciscoPrigogine, I., Stengers, I., (1988) Entre le Temps et l'Eternité, , Fayard, ParisPrigogine, I., Stengers, I., (1984) Order Out of Chaos: Man's New Dialogue with Nature, , Bantam, New YorkCoveney, P., Highfield, R., (1990) The Arrow of Time, , Allen-Ballantine, LondonBogoliubov, N.N., (1962) Studies in Statistical Mechanics I, , J. de Boer and G. E. Uhlenbeck, Eds. (North Holland, Amsterdam)Uhlenbeck, G.E., (1963) Lectures in Statistical Mechanics, , M. Kac, Ed. (Am. Math. Soc., Providence, RI)Buishvili, L.L., Zviadadze, M.D., On the quasi-thermodynamic theory of magnetic relaxation (1972) Physica, 59, p. 697Vasconcellos, A.R., Algarte, A.C., Luzzi, R., On Bogoliubov's hierarchy of relaxation times (1990) Physica A, 166, p. 517Vasconcellos, A.R., Luzzi, R., García-Colín, L.S., Microscopic approach to irreversible thermodynamics I: General theory (1991) Phys. Rev. A, 43, p. 6622Vasconcellos, A.R., Luzzi, R., García-Colín, L.S., Microscopic approach to irreversible thermodynamics II: An example from semiconductor physics (1991) Phys. Rev. A, 43, p. 6633Ramos, J.G., Vasconcellos, A.R., Luzzi, R., A nonequilibrium ensemble formalism: Criterion for truncation of description J. Chem. Phys., , submittedPrigogine, I., The statistical interpretation of nonequilibrium entropy (1973) Acta Physica Austríaca, (SUPPL. X), p. 401Luzzi, R., Ramos, J.G., Vasconcellos, A.R., Rosenfeld-prigogine's complementary principle in the context of informational statistical thermodynamics (1998) Phys. Rev. E, 57, p. 244Hassan, S.A., Vasconcellos, A.R., Luzzi, R., The informational-entropy operator in a nonequilibrium ensemble formalism (1997) IFGW-UNICAMP Internal Report, , Physica A, in pressBogoliubov, N.N., (1970) Lectures in Quantum Statistics II, , Gordon and Breach, New YorkHobson, A., Irreversibility and information in mechanical system (1966) J. Chem. Phys., 45, p. 1352Irreversibility in simple system (1966) Am. J. Phys., 34, p. 411Freidkin, E.S., Nettleton, R.E., Consistency between maximum-entropy formalism and H-theorem (1989) Nuovo Cimento B, 104, p. 597Mayer, J.E., Approach to thermodynamic equilibrium (1961) J. Chem. Phys., 34, p. 1207Lebowitz, J.L., Frisch, H.L., Helfand, E., Nonequilibrium correlation functions in a fluid (1960) Phys. Fluids, 3, p. 325Lewis, R.M., A unifying principle in statistical mechanics (1967) J. Math. Phys., 8, p. 1448Karkheck, J., Van Beijeren, H., Schepper, I.M., Stell, G., Kinetic theory and H-theorem for a dense square-well fluid (1985) Phys. Rev. A, 32, p. 2517Castello, R.C., Martina, E., Löpez De Haro, M., Karkheck, J., Stell, G., Linearized kinetic-variational theory and short-time kinetic theory (1989) Phys. Rev. A, 39, p. 3106Zubarev, D.N., reference [93] see Chapter IV, Section 22Nettleton, R.E., Information-theoretic extended entropy for steady heat conduction in dense fluids (1988) J. Phys. A, 21, p. 3939Nettleton, R.E., Corrections to maximum entropy formalism for steady-state conduction (1989) J. Phys. A, 22, p. 5281Nettleton, R.E., Freidkin, E.S., Nonlinear reciprocity and the maximum entropy formalism (1989) Physica A, 158, p. 672Nettleton, R.E., Evaluation of correlations defining coefficients in nonequilibrium thermodynamics (1992) J. Phys. Soc. Japan, 61, p. 3103Van Kampen, N.G., Chapman-enskog as an application of the method of eliminating fast variables (1987) J. Stat. Phys., 46, p. 709Algarte, A.C., Vasconcellos, A.R., Luzzi, R., Kinetics of hot elementary excitations in photoexcited polar semiconductors (1992) Phys. Stat. Sol. (B), 173, p. 487Vasconcellos, A.R., Algarte, A.C., Luzzi, R., Diffusion of photoinjected carriers in plasma in nonequilibrium polar semiconductors (1993) Phys. Rev. B, 48, p. 10873Luzzi, R., Vasconcellos, A.R., Relaxation processes in nonequilibrium semiconductor plasma (1984) Semiconductor Processes Probed by Ultrafast Laser Spectroscopy, , R. R. Alfano, Ed. (Academic, New York)Vasconcellos, A.R., Luzzi, R., Esperidião, A.S., Propagation of damped hydrodynamic modes in a photoinjected plasma in semiconductors (1995) P