Statistical irreversible thermodynamics in the framework of Zubarev's nonequilibrium statistical operator method

R. Luzzi   ; A. R. Vasconcellos   ; J. G. Ramos   ; C. G. Rodrigues

Teoretičeskaâ i matematičeskaâ fizika, Tome 194 (2018) no. 1, pp. 7-38

Résumé

We describe the formalism of statistical irreversible thermodynamics constructed based on Zubarev's nonequilibrium statistical operator (NSO) method, which is a powerful and universal tool for investigating the most varied physical phenomena. We present brief overviews of the statistical ensemble formalism and statistical irreversible thermodynamics. The first can be constructed either based on a heuristic approach or in the framework of information theory in the Jeffreys–Jaynes scheme of scientific inference; Zubarev and his school used both approaches in formulating the NSO method. We describe the main characteristics of statistical irreversible thermodynamics and discuss some particular considerations of several authors. We briefly describe how Rosenfeld, Bohr, and Prigogine proposed to derive a thermodynamic uncertainty principle.

Détail

Keywords: irreversible thermodynamics, information theory, statistical mechanics.

Bibliographie

[1] \parskip 0.3pt P. W. Anderson, “More is different. Broken symmetry and the nature of the hierarchical structure of science”, Science, 177 (1972), 393–396 | DOI

[2] P. W. Anderson, “Is Complexity Physics? Is it Science? What is it?”, Phys. Today, 44:7 (1991), 9–10 | DOI

[3] R. Luzzi, A. R. Vasconcellos, “Complex behavior in condensed matter: morphological order in dissipative carrier system”, Complexity, 2:5 (1997), 42–49

[4] M. V. Mesquita, A. R. Vasconcellos, R. Luzzi, “Complexity in biological systems”, Contemp. Phys., 40:4 (1999), 247–256 | DOI

[5] G. Nikolis, I. Prigozhin, Poznanie slozhnogo. Vvedenie, Editorial URSS, 2013

[6] R. Luzzi, A. R. Vasconcellos, J. G. Ramos, Statistical Foundations of Irreversible Thermodynamics, Teubner-Texte zur Physik [Teubner Texts in Physics], 35, Teubner, Stuttgart, 2000 | DOI | MR | Zbl

[7] P. T. Landsberg, “Foundations of thermodynamics”, Rev. Modern Phys., 28:4 (1956), 363–392 | DOI | MR | Zbl

[8] A. Hobson, “Irreversibility and information in mechanical systems”, J. Chem. Phys., 45:4 (1966), 1352–1357 | DOI

[9] B. C. Eu, Kinetic Theory of Irreversible Thermodynamics, Wiley, New York, 1992

[10] R. V. Velasco, L. S. Garcia-Colin, “The kinetic foundations of non-local nonequilibrium thermodynamics”, J. Non-Equilib. Thermodyn., 18:2 (1993), 157 | DOI

[11] R. E. Nettleton, “Generalized Grad-type foundations for nonlinear extended thermodynamics”, Phys. Rev. A, 42:8 (1990), 4622–4629 | DOI

[12] L. S. García-Colín, M. López de Haro, R. F. Rodríguez, J. Casas-Vázquez, D. Jou, “On the foundations of extended irreversible thermodynamics”, J. Statist. Phys., 37:3–4 (1984), 465–484 | DOI | MR

[13] D. N. Zubarev, Neravnovesnaya statisticheskaya termodinamika, Nauka, M., 1971 | MR

[14] D. N. Zubarev, “The method of the non-equilibrium statistical operator and its application. I. The non-equilibrium statistical operator”, Fortschr. Physik, 18 (1970), 125–147 | DOI | MR

[15] D. N. Zubarev, “Sovremennye metody statisticheskoi teorii neravnovesnykh protsessov”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem., 15, VINITI, M., 1980, 131–226 | DOI | MR | Zbl

[16] D. N. Zubarev, “Modern methods of the statistical theory of nonequilibrium processes”, J. Soviet Math., 16:6 (1981), 1509–1571 | DOI

[17] R. Luzzi, Á. R. Vasconcellos, J. Galväo Ramos, Predictive Statistical Mechanics. A Nonequilibrium Statistical Ensemble Formalism, Fundamental Theories of Physics, 122, Kluwer, Dordrecht, 2002 | MR

[18] D. N. Zubarev, V. G. Morozov, G. Repke, Statisticheskaya mekhanika neravnovesnykh protsessov, v. 1, Fizmatlit, M., 2002 | MR

[19] D. N. Zubarev, V. G. Morozov, G. Repke, Statisticheskaya mekhanika neravnovesnykh protsessov, v. 2, Fizmatlit, M., 2002 | MR

[20] A. I. Akhiezer, S. V. Peletminskii, Metody statisticheskoi fiziki, Nauka, M., 1977 | MR

[21] J. A. McLennan, “The formal statistical theory of transport processes”, Advances in Chemical Physics, 5, ed. I. Prigogine, Academic Press, New York, 1963, 261–317 | DOI

[22] W. T. Grandy, Principles of Statistical Mechanics, v. 1, Equilibrium Theory, Reidel, Dordrecht, 1987

[23] W. T. Grandy, Principles of Statistical Mechanics, v. 2, Nonequilibrium Phenomena, Reidel, Dordrecht, 1988

[24] B. Robertson, “Equations of motion in nonequilibrium statistical mechanics”, Phys. Rev., 144:1 (1996), 151–161 | DOI | MR

[25] J. P. Dougherty, “Foundations of non-equilibrium statistical mechanics”, Philos. Trans. Roy. Soc. London Ser. A, 346:1680 (1994), 259–305 | MR | Zbl

[26] R. Luzzi, A. R. Vasconcellos, “On the nonequilibrium statistical operator method”, Fortschr. Phys., 38:11 (1990), 887–922 | DOI | MR

[27] R. Zwanzig, “Where do we go from here?”, Perspectives in Statistical Mechanics, ed. H. J. Raveché, North Holland, Amsterdam, 1981, 123–124

[28] J. G. Kirkwood, “The statistical mechanical theory of transport processes I. General theory”, J. Chem. Phys., 14 (1946), 180–201 | DOI

[29] M. S. Green, “Markoff random processes and the statistical mechanics of time-dependent phenomena”, J. Chem. Phys., 20 (1952), 1281–1295 | DOI | MR

[30] H. Mori, I. Oppenheim, J. Ross, “Some topics in quantum statistics. The Wigner function and transport theory”, Studies in Statistical Mechanics, v. I, eds. J. de Boer, G. E. Uhlenbeck, North Holland, Amsterdam, 1962, 213–298 | MR

[31] H. Mori, “Transport, collective motion, and Brownian motion”, Prog. Theor. Phys., 33:3 (1965), 423–455 | DOI

[32] R. Zwanzig, “Statistical mechanics of irreversibility”, Lectures in Theoretical Physics, v. 3, eds. W. E. Brittin, B. W. Downs, J. Downs, Wiley-Interscience, New York, 1961, 106–141 | MR

[33] N. N. Bogolyubov, Problemy dinamicheskoi teorii v statisticheskoi fizike, Gostekhizdat, M., L., 1946 | MR | MR

[34] R. Peierls, “Some simple remarks on the basis of transport theory”, Transport Phenomena (Sitges, Barcelona, June 1974), Lecture Notes in Physics, 31, eds. G. Kirczenow, J. Marro, Springer, Berlin, 1974, 1–33 | DOI | MR

[35] U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques”, Rev. Modern Phys., 29 (1957), 74–93 | DOI | MR | Zbl

[36] B. Robertson, “Introduction to field operators in quantum mechanics”, Amer. J. Phys., 41:5 (1973), 678–690 | DOI | MR

[37] R. Feinman, Statisticheskaya mekhanika, Mir, M., 1975 | MR | Zbl

[38] P. L. Taylor, A Quantum Approach to the Solid State, Prentice Hall, Englewood Cliffs, NJ, 1970

[39] J. M. Ziman, Elements of Advanced Quantum Theory, Cambridge Univ. Press, Cambridge, 1969 | MR

[40] J. Klauder, B. Skagerstam, Coherent States, World Sci., Singapore, 1984 | MR

[41] N. Hugenholtz, “Applications of field theoretical methods to many-boson systems”, Cargèse Lectures in Theoretical Physics, Lecture III, ed. M. Lévy, Benjamin, New York, 1963, 31 pp. | MR

[42] N. S. Krylov, Raboty po obosnovaniyu statisticheskoi fiziki, Iz-vo AN SSSR, M., 1950 | MR

[43] N. N. Bogoliubov, Lectures in Quantum Mechanics, v. 1, Gordon and Breach, New York, 1967 | MR

[44] N. N. Bogoliubov, Lectures in Quantum Mechanics, v. 2, Gordon and Breach, New York, 1970

[45] V. S. Vladimirov, A. A. Logunov, “Pamyati Nikolaya Nikolaevicha Bogolyubova (21.08.1909 – 13.03.1992)”, TMF, 92:2 (1992), 179–181 | DOI | MR

[46] L. Lauck, Á. R. Vasconcellos, R. Luzzi, “A nonlinear quantum transport theory”, Phys. A, 168:2 (1990), 789–819 | DOI

[47] J. R. Madureira, Á. R. Vasconcellos, R. Luzzi, L. Lauck, “Markovian kinetic equations in a nonequilibrium statistical ensemble formalism”, Phys. Rev. E, 57:3 (1998), 3637–3640 | DOI

[48] J. R. Madureira, A. R. Vasconcellos, R. Luzzi, J. Casas-Vazquez, D. Jou, “Evolution of dissipative processes via a statistical thermodynamic approach. I. Generalized Mori–Heisenberg–Langevin equations”, J. Chem. Phys., 108:18 (1998), 7568–7579 | DOI

[49] J. G. Ramos, A. R. Vasconcellos, R. Luzzi, “A classical approach in predictive statistical mechanics: a generalized Boltzmann formalism”, Fortschr. Phys., 43:4 (1995), 265–300 | DOI | MR | Zbl

[50] F. S. Vannucchi, Á. R. Vasconcellos, R. Luzzi, “Thermo-statistical theory of kinetic and relaxation processes”, Internat. J. Modern Phys. B, 23:27 (2009), 5283–5305 | DOI | Zbl

[51] B. Robertson, “Equations of motion in nonequilibrium statistical mechanics. II. Energy transport”, Phys. Rev., 160:1 (1967), 175–183 ; Erratum 166:1 (1968), 206 | DOI | DOI

[52] C. A. B. Silva, Á. R. Vasconcellos, J. G. Ramos, R. Luzzi, “Generalized kinetic equations for far-from-equilibrium many-body systems”, J. Statist. Phys., 143:5 (2011), 1020–1034 | DOI | MR

[53] H. Spohn, “Kinetic equations from Hamiltonian dynamics: Markovian limits”, Rev. Modern Phys., 52:3 (1980), 569–615 | DOI | MR

[54] Yu. L. Klimontovich, Statisticheskaya teoriya otkrytykh sistem, TOO “Yanus”, M., 1995 | Zbl

[55] W. M. Elsasser, “On quantum measurements and the role of the uncertainty relations in statistical mechanics”, Phys. Rev., 52:9 (1937), 987–999 | DOI

[56] E. T. Jaynes, “A backward look to the future”, Physics and Probability, eds. W. T. Grandy, P. W. Milonni, Cambridge Univ. Press, Cambridge, 1993, 261–275

[57] E. T. Jaynes, “Notes on present status and future prospects”, Maximum Entropy and Bayesian Methods (Laramie, Wyoming, 1990), Fundamental Theories of Physics, 43, eds. W. T. Grandy, L. H. Schick, Springer, Netherlands, 1991, 1–13 | DOI | MR

[58] E. T. Jaynes, Probability Theory: The Logic of Science, Cambridge Univ. Press, Cambridge, 2002 | MR

[59] E. B. Davies, “Markovian master equations”, Commun. Math. Phys., 39:2 (1994), 91–110 | DOI | MR

[60] R. Kurant, D. Gilbert, Metody matematicheskoi fiziki, Gosudarstvennoe izdatelstvo tekhniko-teoreticheskoi literatury, M., L., 1951 | MR | MR | Zbl

[61] H. Barnum, C. M. Caves, C. Fuchs, R. Schack, D. J. Driebe, W. G. Hoover, H. Posch, B. L. Holian, R. Peierls, J. L. Lebowitz, “Is Boltzmann entropy time's arrow's archer?”, Phys. Today, 47:11 (1994), 11–15 | DOI

[62] H. B. Callen, Thermodynamics. An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics, John Wiley and Sons, New York | Zbl

[63] S. R. De Groot, P. Mazur, Non-equilibrium Thermodynamics, North Holland, Amsterdam, 1962 | MR

[64] L. Onsager, “Reciprocal relations in irreversible processes. I”, Phys. Rev., 37:4 (1931), 405–426 | DOI | Zbl

[65] L. Onsager, “Reciprocal relations in irreversible processes. II”, Phys. Rev., 38:12 (1931), 2265–2279 | DOI | Zbl

[66] P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, New York, 1971 | Zbl

[67] L. Tisza, “Concluding remarks”, Thermodynamics: History and Philosophy. Facts, Trends, Debates (Veszprém, Hungary, July 23–28, 1990), eds. K. Martinás, L. Ropolyi, P. Szegedi, World Sci., Singapore, 1991, 515–522

[68] C. Truesdell, Rational Thermodynamics, Springer, Berlin, 1988 | MR

[69] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, Springer, Berlin, 2010 | MR

[70] D. Jou, J. Casas-Vázquez, G. Lebon, “Extended irreversible thermodynamics”, Rep. Prog. Phys., 51:8 (1988), 1105–1179 | DOI | MR

[71] G. Lebon, D. Jou, J. Casas-Vázquez, “Questions and answers about a thermodynamic theory of the third type”, Contemp. Phys., 33:1 (1992), 41–51 | DOI

[72] G. Lebon, D. Jou, “Early history of extended irreversible thermodynamics (1953–1983): an exploration beyond local equilibrium and classical transport theory”, Eur. J. Phys. H, 40:2 (2015), 205–240 | DOI

[73] D. Jou, J. Casas-Vázquez, G. Lebon, “Extended irreversible thermodynamics revisited”, Rep. Progr. Phys., 62:7 (1999), 1035–1142 | DOI | MR

[74] I. Gyarmati, “On the wave appproach to thermodynamics and some problems of non-linear theories”, J. Non-Equilib. Thermodyn., 2:4, 233–260 | DOI

[75] M. Grmela, “Thermodynamics of driven systems”, Phys. Rev. E, 48:2 (1993), 919–930 | DOI | MR

[76] N. Bernardes, “Thermodynamics and complementarity”, Phys. A, 260:1–2 (1998), 186–200 | DOI

[77] Á. R. Vasconcellos, R. Luzzi, J. G. Ramos, “Irreversible thermodynamics in a nonequilibrium statistical ensemble formalism”, La Rivista del Nuovo Cimento, 24:3 (2001), 1–70

[78] R. Luzzi, Á. R. Vasconcellos, J. G. Ramos, “The theory of irreversible processes: foundations of a non-equilibrium statistical ensemble formalism”, La Rivista del Nuovo Cimento, 29:2 (2006), 1–82 | DOI

[79] R. Luzzi, Á. R. Vasconcellos, J. G. Ramos, “Non-equilibrium statistical mechanics of complex systems: an overview”, La Rivista del Nuovo Cimento, 30:3 (2007), 95–157 | DOI

[80] C. A. B. Silva, J. G. Ramos, Á. R. Vasconcellos, R. Luzzi, Nonlinear higher-order hydrodynamics. Unification of kinetic and hydrodynamic approaches within a nonequilibrium statistical ensemble formalism, arXiv: 1210.7280

[81] C. G. Rodrigues, Á. R. Vasconcellos, R. Luzzi, “Mesoscopic hydro-thermodynamics of phonons in semiconductors: heat transfer in III-nitrides”, Eur. Phys. J. B, 86:5 (2013), 200, 9 pp. | DOI | MR

[82] Á. R. Vasconcellos, A. R. B. de Castro, C. A. B. Silva, R. Luzzi, “Mesoscopic hydro-thermodynamics of phonons”, AIP Adv., 3:7 (2013), 072106–072133 | DOI

[83] C. A. B. Silva, C. G. Rodrigues, J. G. Ramos, R. Luzzi, “Higher-order generalized hydrodynamics: foundations within a nonequilibrium statistical ensemble formalism”, Phys. Rev. E, 91:6 (2015), 063011, 15 pp. | DOI | MR

[84] C. G. Rodrigues, A. R. B. Castro, R. Luzzi, “Higher-order generalized hydrodynamics of carriers and phonons in semiconductors in the presence of electric fields: macro to nano”, Phys. Stat. Sol. B, 252:12 (2015), 2802–2819 | DOI

[85] C. G. Rodrigues, Á. R. Vasconcellos, R. Luzzi, “Thermal conductivity in higher-order generalized hydrodynamics: characterization of nanowires of silicon and gallium nitride”, Phys. E, 60 (2014), 50–58 | DOI

[86] J. L. del Río, L. S. García-Colín, “Repeated randomness assumption and the projection operator formalism”, Phys. Rev. E, 54:1 (1996), 950–953 | DOI | MR

[87] C. E. Shannon, W. Weaver, The Mathematical Theory of Communication, Univ. Illinois Press, Urbana, IL, 1949 | MR

[88] L. Brillouin, Science and Information Theory, Academic Press, New York, 1962 | MR

[89] I. Prigogine, Étude Thermodinamique des Phénomenès Irreversibles, Dover, Liège, 1947

[90] S. A. Hassan, A. R. Vasconcellos, R. Luzzi, “Informational-statistical thermodynamics of a dissipative system in a steady state”, Physica A, 235:3–4 (1997), 345–368 | DOI

[91] E. T. Jaynes, “The evolution of Carnot's principle”, Maximum Entropy and Bayesian Methods in Science and Engineering (Laramie, Wyoming, August 5–8, 1985), eds. G. J. Erickson, C. R. Smith, Kluwer, Dordrecht, 1988, 267–281 | MR

[92] L. Rosenfeld, “On the foundations of statistical thermodynamics”, Acta Phys. Polon., 14 (1955), 3–29 | MR

[93] L. Rosenfeld, “Questions on irreversibility and ergodicity”, Proceedings of the International School of Physics. “Enrico Fermi”, Course XIV, ed. P. Caldirola, Academic Press, New York, 1960, 1–20 | MR

[94] R. Luzzi, J. G. Ramos, Á. R. Vasconcellos, “Rosenfed–Prigogine complementarity of descriptions in the context of informational statistical thermodynamics”, Phys. Rev. E, 57:1 (1998), 244–251 | DOI

[95] E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics, Synthese Library, 158, Reidel, Dordrecht, 1983 | MR

[96] E. T. Jaynes, “Gibbs vs Boltzmann entropies”, Amer. J. Phys., 33:5 (1965), 391–399 | DOI

[97] G. Nikolis, I. Prigozhin, Samoorganizatsiya v neravnovesnykh sistemakh: ot dissipativnykh struktur k uporyadochennosti cherez fluktuatsii, Mir, M., 1979 | MR

[98] G. Nicolis, “Dissipative systems”, Rep. Progr. Phys., 49:8 (1986), 873–949 | DOI | MR

[99] D. Jou, J. Casas-Vázquez, “Possible experiment to check the reality of a nonequilibrium temperature”, Phys. Rev. A, 45:12 (1992), 8371–8373 | DOI

[100] R. Luzzi, Á. R. Vasconcellos, J. Casas-Vázquez, D. Jou, “On the selection of the state space in nonequilibrium thermodynamics”, Phys. A, 248:1–2 (1998), 111–137 | DOI

[101] R. Luzzi, Á. R. Vasconcellos, “Response function theory for far-from-equilibrium systems”, J. Statist. Phys., 23:5 (1980), 539–559 | DOI | MR

[102] Á. R. Vasconcellos, R. Luzzi, D. Jou, J. Casas-Vázquez, “Thermodynamic variables in the context of a nonequilibrium statistical ensemble approach”, J. Chem. Phys., 107:18 (1997), 7383–7396 | DOI

[103] A. C. Algarte, Á. R. Vasconcellos, R. Luzzi, “Kinetics of hot elementary excitations in photoexcited polar semiconductors”, Phys. Stat. Sol. B, 173 (1992), 487–514 | DOI

[104] A. C. Algarte, A. R. Vasconcellos, R. Luzzi, “Ultrafast kinetics of evolution of optical phonons in a photoinjected highly excited plasma in semiconductors”, Phys. Rev. B, 54:16 (1996), 11311–11316 | DOI

[105] A. C. Algarte, A. R. Vasconcellos, R. Luzzi, “Ultrafast phenomena in the photoinjected plasma in semiconductors”, Braz. J. Phys., 26 (1996), 543–552

[106] A. C. Algarte, A. R. Vasconcellos, R. Luzzi, “Cooling of hot carriers in highly photoexcited semiconductors”, Phys. Rev. B, 38:3 (1988), 2162–2165 | DOI

[107] N. Bohr, “On the notions of causality and complementarity”, Dialectica, 2:3–4 (1948), 312–319 | DOI | Zbl

[108] I. Prigozhin, Ot suschestvuyuschego k voznikayuschemu. Vremya i slozhnost v fizicheskikh naukakh, Nauka, M., 2002

[109] H. Atlan, Entre le cristal et la fumèe. Essai sur l'organisation du vivant, Seuil, Paris, 1986

[110] I. Prigogine, I. Stengers, Order out of Chaos, Bantam, New York, 1984

[111] R. Landauer, “Information is physical”, Phys. Today, 44:5 (1991), 23–31 | DOI

[112] S. J. Kline, N. Rosenberg, “An overview of innovation”, The Positive Sum Strategy: Harnessing Technology for Economic Growth, eds. R. Landau, N. Rosenberg, National Acad. Sci., Washington, DC, 1986, 275–305 | DOI

[113] E. Lutz, S. Ciliberto, “Information: From Maxwell's demon to Landauer's eraser”, Phys. Today, 68:9 (2015), 30–37 | DOI

[114] L. Szilard, “Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen”, Z. Physik, 53:11–12 (1929), 840–856 | DOI | Zbl

[115] C. H. Bennett, “The thermodynamics of computation – a review”, Internat. J. Theor. Phys., 21:12 (1982), 905–940 | DOI

[116] L. Sklar, Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics, Cambridge Univ. Press, Cambridge, 1993 | MR

[117] J. Bricmont, D. Dürr, M. C. Galavotti, G. Ghirardi, F. Petruccione, N. Zanghi, Chance in Physics: Foundations and Perspectives, Lecture Notes in Physics, 574, Springer, Heidelberg, 2001 | DOI

[118] J. Bricmont, “Science of chaos or chaos in science?”, Physicalia Magazine, 17:3–4 (1995), 159–208

[119] J. Bricmont, “Science of chaos or chaos in science?”, Ann. New York Academy of Sciences, 775 (1996), 131–175 | DOI

[120] J. Meixner, “The entropy problem in thermodynamic processes”, Rheologica Acta, 12:3 (1973), 465–467 | DOI

[121] J. Meixner, “Entropy and entropy production”, Foundations of Continuum Thermodynamics, eds. J. J. Delgado, M. N. Nina, J. H. Whitelaw, MacMillan, London, 1974, 129–141

[122] J. Meixner, H. G. Reik, “Thermodynamik der irreversiblen Prozesse”, Principles of Thermodynamics and Statistics. Handbuch der Physik, v. 3, ed. S. Flügge, Springer, Berlin, 1959, 413–523 | DOI | MR

[123] S. Abe, Y. Okamoto (eds.), Nonextensive Statistical Mechanics and its Applications (Okazaki, February 15–18, 1999), Lecture Notes in Physics, 560, Springer, Berlin, 2001 | DOI | MR

[124] J. N. Kapur, H. K. Kesavan, Entropy Optimization Principles with Applications, Academic Press, Boston, 1992 | MR

[125] C. E. Shannon, “A mathematical theory of communication I”, Bell Syst. Tech. J., 27:3 (1948), 379–423 | DOI | MR | Zbl

[126] C. E. Shannon, “A mathematical theory of communication II”, Bell Syst. Tech. J., 27:4 (1948), 623–656 | DOI | MR

[127] N. J. A. Sloane, A. D. Wyner (eds.), Claude Elwood Shannon. Collected Papers, Wiley-Interscience, New York, 1993 | MR | Zbl

[128] R. T. Cox, The Algebra of Probable Inference, The Johns Hopkins Univ. Press, Baltimore, 1961 | MR | Zbl

[129] A. Cho, “A fresh take on disorder, or disorderly science?”, Science, 297:5585 (2002), 1268–1269 | DOI

[130] R. Luzzi, Á. R. Vasconcellos, J. G. Ramos, “Trying to make sense of disorder”, Science, 298:5596 (2002), 1171–1172 | DOI

[131] R. Luzzi, Á. R. Vasconcellos, J. G. Ramos, “Letter to the Editor: ‘On fallacies concerning nonextensive thermodynamics and q-entropy’”, Europhys. News, 37:2 (2006), 11

[132] R. Balian, M. Nauenberg, “Letter to the Editor”, Europhys. News, 37:2 (2006), 9