Zubarev's nonequilibrium statistical operator method in the generalized statistics of multiparticle systems

P. A. Glushak   ; B. B. Markiv   ; M. V. Tokarchuk

Teoretičeskaâ i matematičeskaâ fizika, Tome 194 (2018) no. 1, pp. 71-89

Résumé

We present a generalization of Zubarev's nonequilibrium statistical operator method based on the principle of maximum Renyi entropy. In the framework of this approach, we obtain transport equations for the basic set of parameters of the reduced description of nonequilibrium processes in a classical system of interacting particles using Liouville equations with fractional derivatives. For a classical systems of particles in a medium with a fractal structure, we obtain a non-Markovian diffusion equation with fractional spatial derivatives. For a concrete model of the frequency dependence of a memory function, we obtain generalized Kettano-type diffusion equation with the spatial and temporal fractality taken into account. We present a generalization of nonequilibrium thermofield dynamics in Zubarev's nonequilibrium statistical operator method in the framework of Renyi statistics.

Détail

Keywords: Renyi entropy, nonequilibrium statistical operator, generalized transport equation
Mots-clés : diffusion equation.

Bibliographie

[1] D. N. Zubarev, “Vychislenie konfiguratsionnogo integrala sistemy chastits s kulonovskim vzaimodeistviem”, Dokl. AN SSSR, 95:4 (1954), 757–760 | MR | Zbl

[2] D. N. Zubarev, “Dvukhvremennye funktsii Grina v statisticheskoi fizike”, UFN, 71:1 (1960), 71–116 | DOI | DOI

[3] D. N. Zubarev, “Statisticheskii operator dlya neravnovesnykh sistem”, Dokl. AN SSSR, 140:1 (1961), 92–95 | Zbl

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

[5] 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

[6] D. N. Zubarev, “Nonequilibrium statistical operator as a generalization of Gibbs distribution for nonequilibrium case”, Condens. Matter Phys., 1994, no. 4, 7–25 | DOI

[7] V. G. Morozov, G. Röpke, “Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes”, Condens. Matter Phys., 1:4(16) (1998), 673–686 | DOI

[8] D. N. Zubarev, V. G. Morozov, G. Repke, Statisticheskaya mekhanika neravnovesnykh protsessov, v. 1, Fizmatlit, M., 2002; D. N. Zubarev, V. G. Morozov, G. Röpke, Statistical Mechanics of Nonequilibrium Processes, т. 1, Basic Concepts, Kinetic Theory, Akademie, Berlin, 1996 | MR

[9] D. N. Zubarev, V. G. Morozov, G. Repke, Statisticheskaya mekhanika neravnovesnykh protsessov, v. 2, Fizmatlit, M., 2002; D. N. Zubarev, V. G. Morozov, G. Röpke, Statistical Mechanics of Nonequilibrium Processes, т. 2, Relaxation and Hydrodynamic Processes, Akademie, Berlin, 1997 | MR

[10] I. R. Yukhnovskii, M. F. Golovko, Statisticheskaya teoriya klassichekikh ravnovesnykh sistem, Naukova dumka, Kiev, 1980 | MR

[11] I. R. Yukhnovskii, “Metod kollektivnykh peremennykh s sistemoi otscheta dlya bolshogo kanonicheskogo ansamblya”, TMF, 79:2 (1989), 282–296 | DOI

[12] I. R. Yukhnovskii, O. V. Patsahan, “Grand canonical distribution for multicomponent system in the collective variables method”, J. Statist. Phys., 81:3–4 (1995), 647–672 | DOI | MR | Zbl

[13] I. R. Yukhnovskii, Vibrani pratsi. Fizika, Vid. Natsionalnogo universitetu “Lvivska politekhnika”, Lviv, 2005

[14] I. R. Yukhnovskii, Phase Transitions of the Second Order. Collective Variables Method, World Sci., Singapore, 1987 | MR

[15] I. R. Yukhnovskii, “The functional of the grand partition function for the investigation of the liquid-gas critical point”, Phys. A, 168:3 (1990), 999–1020 | DOI

[16] I. R. Yukhnovskii, M. P. Kozlovskii, I. V. Pilyuk, Mikroskopichna teoriya fazovikh perekhodiv u trivimirnikh sistemakh, {\fontencoding{X2}\selectfontЄvrosvit, Lviv, 2001

[17] I. A. Vakarchuk, I. R. Yukhnovskii, “Samosoglasovannoe opisanie dalnodeistvuyuschikh i korotkodeistvuyuschikh korrelyatsii v teorii zhidkogo $\operatorname{He}^4$. I”, TMF, 40:1 (1979), 100–111 | DOI

[18] I. A. Vakarchuk, O. L. Gonopolskii, I. R. Yukhnovskii, “Samosoglasovannoe opisanie dalnodeistvuyuschikh i korotkodeistvuyuschikh korrelyatsii v teorii zhidkogo $\operatorname{He}^4$. II”, TMF, 41:1 (1979), 77–88 | DOI

[19] P. P. Kostrobii, I. R. Yukhnovskii, “Funktsii raspredeleniya vyrozhdennogo elektronnogo gaza v periodicheskom vneshnem pole”, TMF, 32:2 (1977), 208–222 | DOI

[20] V. L. Bonch-Bruevich, S. V. Tyablikov, Metod funktsii Grina v statisticheskoi mekhanike, Fizmatlit, M., 1961 | MR

[21] N. M. Plakida, “Metod dvukhvremennykh funktsii Grina v teorii angarmonicheskikh kristallov”, Statisticheskaya fizika i kvantovaya teoriya polya, ed. N. N. Bogolyubov, Nauka, M., 1973, 205–240 | MR

[22] N. M. Plakida, Nekotorye voprosy kvantovoi teorii tverdogo tela (metod dvukhvremennykh funktsii Grina), Izd-vo Mosk. un-ta, M., 1974

[23] D. N. Zubarev, Yu. A. Tserkovnikov, “Metod dvukhvremennykh temperaturnykh funktsii Grina v ravnovesnoi i neravnovesnoi statisticheskoi mekhanike”, Tr. MIAN, 175 (1986), 134–177 | MR | Zbl

[24] N. M. Plakida, “Dvukhvremennye funktsii Grina v teorii sverkhprovodimosti”, TMF, 154:1 (2008), 129–146 | DOI | DOI | Zbl

[25] N. M. Plakida, “Dvukhvremennye funktsii Grina i diagrammnaya tekhnika”, TMF, 168:3 (2011), 518–535 | DOI | DOI | Zbl

[26] V. D. Kreft, D. Kremp, V. Ebeling, G. Repke, Kvantovaya statistika sistem zaryazhennykh chastits, Mir, M., 1988

[27] G. D. Mahan, Many-Particle Physics, Kluwer, New York, 2000 | MR

[28] V. T. Shvets, Metod funktsii Grina v teoriïmetaliv, Latstar, Odessa, 2002

[29] R. Luzzi, A. R. Vasconcellos, J. G. Ramos, Predictive Statistical Mechanics. A Nonequilibrium Ensemble Formalism, Fundamental Theories of Physics, 122, Springer Science Business Media, Dordrecht, 2002 | MR

[30] Yu. A. Tserkovnikov, Statisticheskaya mekhanika. Izbrannye trudy, Yanus-K, M., 2010

[31] I. V. Stasyuk, Funktsiï Grina y kvantovii statistitsi tverdikh til: posibnik, Lvivskii natsionalnii universitet imeni Ivana Franko, Lviv, 2013

[32] R. Luzzi, A. R. Vasconcellos, J. G. Ramos, Statistical Foundations of Irreversible Thermodynamics, Springer Science and Business Media, Berlin, 2013 | MR

[33] I. I. Lyapilin, V. P. Kalashnikov, Neravnovesnyi statisticheskii operator i ego prilozheniya k kinetike paramagnitnykh yavlenii v provodyaschikh kristallakh, UrO RAN, Ekaterinburg, 2008

[34] G. Röpke, Nonequilibrium Statistical Physics, Wiley-VCH, New York, 2013 | MR | MR

[35] A. A. Khamzin, R. R. Nigmatulin, Metod neravnovesnogo statisticheskogo operatora i ego prilozhenie k kinetike izingovskikh magnetikov, Uchebnoe posobie, Kazan. un-t, Kazan, 2011

[36] I. M. Mryglod, M. V. Tokarchuk, “K statisticheskoi gidrodinamike prostykh zhidkostei”, Voprosy atomnoi nauki i tekhniki, 3(24) (1992), 134–139

[37] I. M. Mryglod, I. P. Omelyan, M. V. Tokarchuk, “Generalized collective modes for the Lennard-Jones fluid”, Mol. Phys., 84:2 (1995), 235–259 | DOI

[38] B. B. Markiv, I. P. Omelyan, M. V. Tokarchuk, “Neravnovesnyi statisticheskii operator v obobschennoi molekulyarnoi gidrodinamike zhidkostei”, TMF, 154:1 (2008), 91–101 | DOI | DOI | MR | Zbl

[39] B. B. Markiv, I. P. Omelyan, M. V. Tokarchuk, “Relaxation to the state of molecular hydrodynamics in the generalized hydrodynamics of liquids”, Phys. Rev. E, 82:4 (2010), 041202, 11 pp. | DOI

[40] I. M. Mryglod, “Generalized statistical hydrodynamics of fluids: approach of generalized collective modes”, Condens. Matter Phys., 1:4(16) (1998), 753–796 | DOI

[41] I. P. Omelyan, I. M. Mryglod, M. V. Tokarchuk, “Dielectric relaxation in dipolar fluids. Generalized mode approach”, Condens. Matter Phys., 1:1(13) (1998), 179–200 | DOI

[42] I. P. Omelyan, I. M. Mryglod, M. V. Tokarchuk, “Generalized dipolar modes of a Stockmayer fluid in high-order approximations”, Phys. Rev. E., 57:6 (1998), 6667–6676 | DOI

[43] I. M. Mryglod, M. V. Tokarchuk, R. Folk, “On the hydrodynamic theory of a magnetic liquid I. General description”, Phys. A, 220:3–4 (1995), 325–348 | DOI

[44] I. M. Mryglod, M. V. Tokarchuk, “Statisticheskaya gidrodinamika magnitnykh zhidkostei. I. Metod neravnovesnogo statisticheskogo operatora”, TMF, 115:1 (1998), 132–153 | DOI | DOI | Zbl

[45] D. N. Zubarev, M. V. Tokarchuk, “Neravnovesnaya statisticheskaya gidrodinamika ionnykh sistem”, TMF, 70:2 (1987), 234–254 | DOI

[46] B. Markiv, A. Vasylenko, M. Tokarchuk, “Statistical description of hydrodynamic processes in ionic melts while taking into account polarization effects”, J. Chem. Phys., 136:23 (2012), 234502, 10 pp. | DOI

[47] B. Markiv, M. Tokarchuk, “Effect of ion polarization on longitudinal excitations in ionic melts”, J. Chem. Phys., 143:19 (2015), 194509, 8 pp. | DOI

[48] M. V. Tokarchuk, “K statisticheskoi teorii neravnovesnoi plazmy v sobstvennom elektromagnitnom pole”, TMF, 97:1 (1993), 27–43 | DOI

[49] B. Markiv, M. Tokarchuk, “Consistent description of kinetics and hydrodynamics of dusty plasma”, Phys. Plasmas, 21:2 (2014), 023707, 16 pp. | DOI

[50] V. V. Ignatyuk, I. M. Mryglod, M. V. Tokarchuk, “K teopii dinamicheskikh svoistv polukvantovogo geliya”, FNT, 25:5 (1999), 407–416 | DOI

[51] V. V. Ignatyuk, I. M. Mryglod, M. V. Tokarchuk, “Vremennye korrelyatsionnye funktsii i obobschennye koeffitsienty perenosa polukvantovogo geliya”, FNT, 25:11 (1999), 1145–1153 | DOI

[52] P. P. Kostrobii, M. V. Tokarchuk, B. M. Markovich, V. V. Ignatyuk, B. V. Gnativ, Reaktsiino-difuziini protsesi v sistemakh “metal–gaz”, Vidav. NU “Lvivska politekhnika”, Lviv, 2009

[53] D. N. Zubarev, A. V. Prozorkevich, S. A. Smolyanskii, “Vyvod nelineinykh obobschennykh uravnenii kvantovoi relyativistskoi gidrodinamiki”, TMF, 40:3 (1979), 394–407 | MR

[54] A. V. Prozorkevich, V. L. Samorodov, S. A. Smolyanskii, “Kvantovaya relyativistskaya gidrodinamika sistem s narushennoi simmetriei. I. Lokalno-ravnovesnoe sostoyanie”, TMF, 52:3 (1982), 463–472 | DOI | Zbl

[55] S. A. Smolyanskii, “Kvantovaya relyativistskaya gidrodinamika sistem s narushennoi simmetriei. II. Neravnovesnoe sostoyanie”, TMF, 52:2 (1982), 292–300 | DOI

[56] F. Becattini, L. Tinti, “Nonequilibrium thermodynamical inequivalence of quantum stress-energy and spin tensors”, Phys. Rev. D, 87:2 (2013), 025029, 15 pp. | DOI

[57] L. Tinti, “Thermodynamical inequivalence of stress-energy and spin tensors”, Progress in Mathematical Relativity, Gravitation and Cosmology, Proceedings of the Spanish Relativity Meeting ERE2012 (University of Minho, Guimaraes, Portugal, September 3–7, 2012), eds. A. García-Parrado, F. C. Mena, F. Moura, E. Vaz, Springer, Berlin, Heidelberg, 2014, 433–437

[58] F. Becattini, L. Bucciantini, E. Grossi, L. Tinti, “Local thermodynamical equilibrium and the $\beta$ frame for a quantum relativistic fluid”, Eur. Phys. J. C, 75:5 (2015), 191, 17 pp. | DOI

[59] D. N. Zubarev, M. V. Tokarchuk, “Neravnovesnaya termopolevaya dinamika i metod neravnovesnogo statisticheskogo operatora. I. Osnovnye sootnosheniya”, TMF, 88:2 (1991), 286–310 | DOI | MR

[60] M. V. Tokarchuk, T. Arimitsu, A. E. Kobryn, “Thermo field hydrodynamic and kinetic equations of dense quantum nuclear systems”, Condens. Matter Phys., 1:3(15) (1998), 605–642 | DOI

[61] D. N. Zubarev, V. N. Klimov, “K teorii temperaturnogo skachka na granitse plazmy v magnitnom pole”, Fizika plazmy i problema upravlyaemykh termoyadernykh reaktsii, v. 1, Izd-vo AN SSSR, M., 1958, 138–160

[62] D. N. Zubarev, V. N. Klimov, “Statsionarnye rezhimy magnitnogo termoyadernogo reaktora”, Fizika plazmy i problema upravlyaemykh termoyadernykh reaktsii, v. 1, Izd-vo AN SSSR, M., 1958, 249–288

[63] A. V. Nedospasov, “Fizika pristenochnoi plazmy v tokamakakh”, UFN, 152:3 (1987), 479–492 | DOI | DOI

[64] A. V. Nedospasov, M. Z. Tokar, “Pristenochnaya plazma v tokamakakh”, Voprosy teorii plazmy. Vyp. 18, ed. B. B. Kadomtsev, Energoatomizdat, M., 1990, 68–208

[65] V. N. Tsytovich, “Plazmenno-pylevye kristally, kapli i oblaka”, UFN, 167:1 (1997), 57–99 | DOI | DOI

[66] V. N. Tsytovich, D. Vinter, “Pyl v ustanovkakh upravlyaemogo termoyadernogo sinteza”, UFN, 168:8 (1998), 899–907 | DOI | DOI

[67] V. N. Tsytovich, “O perspektivakh eksperimentalnykh i teoreticheskikh issledovanii samoorganizovannykh pylevykh struktur v kompleksnoi plazme v usloviyakh mikrogravitatsii”, UFN, 185:2 (2015), 161–179 | DOI | DOI

[68] V. E. Fortov, A. G. Khrapak, S. A. Khrapak, V. I. Molotkov, O. F. Petrov, “Pylevaya plazma”, UFN, 174:5 (2004), 495–544 | DOI | DOI

[69] D. N. Zubarev, V. G. Morozov, I. P. Omelyan, M. V. Tokarchuk, “O kineticheskikh uravneniyakh dlya plotnykh gazov i zhidkostei”, TMF, 87:1 (1991), 113–129 | DOI | MR | Zbl

[70] D. N. Zubarev, V. G. Morozov, I. P. Omelyan, M. V. Tokarchuk, “Ob'edinenie kineticheskogo i gidrodinamicheskogo podkhodov v teorii plotnykh gazov i zhidkostei”, TMF, 96:3 (1993), 325–350 | DOI | MR | Zbl

[71] M. V. Tokarchuk, I. P. Omelyan, A. E. Kobryn, “A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method”, Condens. Matter Phys., 1:4(16) (1998), 687–751 | DOI

[72] A. E. Kobryn, I. P. Omelyan, M. V. Tokarchuk, “The modified group expansions for construction of solutions to the BBGKY hierarchy”, J. Statist. Phys., 92:5–6 (1998), 973–994 | DOI | MR | Zbl

[73] B. Markiv, I. Omelyan, M. Tokarchuk, “Consistent description of kinetics and hydrodynamics of weakly nonequilibrium processes in simple liquids”, J. Statist. Phys., 155:5 (2014), 843–866 | DOI | MR | Zbl

[74] D. N. Zubarev, V. G. Morozov, “Formulirovka granichnykh uslovii k tsepochke Bogolyubova s uchetom lokalnykh zakonov sokhraneniya”, TMF, 60:2 (1984), 270–279 | DOI | MR

[75] V. G. Morozov, A. E. Kobryn, M. V. Tokarchuk, “Modified kinetic theory with consideration for slow hydrodynamical processes”, Condens. Matter Phys., 1994, no. 4, 117–127 | DOI

[76] P. A. Hlushak, M. V. Tokarchuk, “Chain of kinetic equations for the distribution functions of particles in simple liquid taking into account nonlinear hydrodynamic fluctuations”, Phys. A, 443 (2016), 231–245 | DOI | MR

[77] I. R. Yukhnovskii, P. A. Hlushak, M. V. Tokarchuk, “BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids”, Condens. Matter Phys., 19:4 (2016), 43705, 18 pp. | DOI

[78] D. N. Zubarev, “Statisticheskaya termodinamika protsessov turbulentnogo perenosa”, TMF, 53:1 (1982), 93–107 | DOI | MR

[79] B. B. Markiv, R. M. Tokarchuk, P. P. Kostrobij, M. V. Tokarchuk, “Nonequilibrium statistical operator method in Renyi statistics”, Phys. A, 390:5 (2011), 785–791 | DOI | MR

[80] P. Kostrobij, R. Tokarchuk, M. Tokarchuk, B. Markiv, “Zubarev nonequilibrium statistical operator method in Renyi statistics. Reaction-diffusion processes”, Condens. Matter Phys., 17:3 (2014), 33005, 9 pp. | DOI

[81] P. P. Kostrobii, A. V. Viznovich, B. B. Markiv, M. V. Tokarchuk, “Obobschennye kineticheskie uravneniya dlya plotnykh gazov i zhidkostei v metode neravnovesnogo statisticheskogo operatora Zubareva i statistike Reni”, TMF, 184:1 (2015), 145–159 | DOI | DOI | MR

[82] A. Rényi, Probability Theory, North-Holland Series in Applied Mathematics and Mechanics, 10, North-Holland, Amsterdam, 1970 | MR

[83] Yu. L. Klimontovich, Vvedenie v fiziku otkrytykh sistem, Yanus-K, M., 2002

[84] A. G. Bashkirov, “Entropiya Reni kak statisticheskaya entropiya dlya slozhnykh sistem”, TMF, 149:2 (2006), 299–317 | DOI | DOI | MR | Zbl

[85] R. Luzzi, A. R. Vasconcellos, J. G. Ramos, “Non-equilibrium statistical mechanics of complex systems: an overview”, Riv. Nuovo Cimento, 30:3 (2007), 95–158 | DOI

[86] A. R. Vasconcellos, J. G. Ramos, A. Gorenstein, M. U. Kleinke, T. G. Souza Cruz, R. Luzzi, “Statistical approach to non-Fickian diffusion”, Internat. J. Modern Phys. B, 20:28 (2006), 4821–4841 | DOI | Zbl

[87] R. R. Nigmatullin, “Drobnyi integral i ego fizicheskaya interpretatsiya”, TMF, 90:3 (1992), 354–368 | DOI | MR | Zbl

[88] C. Tsallis, “Possible generalization of Boltzmann–Gibbs statistics”, J. Statist. Phys., 52:1–2 (1988), 479–487 | DOI | MR | Zbl

[89] C. Tsallis (ed.), Introduction to Nonextensive Statistical Mechanics. Approaching a Complex World, Springer, New York, 2009 | MR

[90] P. Turán (ed.), Selected Papers by Alfréd Rényi, v. 1, 2, Akadémiai Kiadó, Budapest, 1976 | MR

[91] B. D. Sharma, D. P. Mittal, “New nonadditive measures of entropy for discrete probability distributions”, J. Math. Sci., 10 (1975), 28–40 | MR

[92] E. Aktürk, G. B. Bağci, R. Sever, Is Sharma-Mittal entropy really a step beyond Tsallis and Rényi entropies?, arXiv: cond-mat/0703277

[93] C. Beck, “Superstatistics: theory and applications”, Contin. Mech. Thermodyn., 16:3 (2004), 293–304 | DOI | MR | Zbl

[94] C. Beck, “Superstatistics: theoretical concepts and physical applications”, Anomalous Transport: Foundations and Applications, eds. R. Klages, G. Radons, I. M. Sokolov, Wiley-VCH, New York, 2008, 433–457 | DOI

[95] V. V. Uchaikin, Metod drobnykh proizvodnykh, Artishok, Ulyanovsk, 2008

[96] D. Korošak, B. Cvikl, J. Kramer, R. Jecl, A. Prapotnik, “Fractional calculus applied to the analysis of spectral electrical conductivity of clay-water system”, J. Contain. Hydrol., 92:1–2 (2007), 1–9 | DOI

[97] R. Metzler, J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach”, Phys. Rep., 339:1 (2000), 1–77 | DOI | MR | Zbl

[98] J. Bisquert, A. Compte, “Theory of the electrochemical impedance of anomalous diffusion”, J. Electroanal. Chem., 499:1 (2001), 112–120 | DOI

[99] T. Kosztołowicz, K. D. Lewandowska, “Hyperbolic subdiffusive impedance”, J. Phys. A: Math. Theor., 42:5 (2009), 055004, 14 pp. | DOI | MR | Zbl

[100] J.-P. Bouchaud, A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications”, Phys. Rep., 195:4–5 (1990), 127–293 | DOI | MR

[101] R. R. Nigmatullin, “To the theoretical explanation of the ‘Universal response’”, Phys. Stat. Sol. (b), 123:2 (1984), 739–745 | DOI

[102] R. R. Nigmatullin, “On the theory of relaxation for systems with ‘remnant’ memory”, Phys. Stat. Sol. (b), 124:1 (1984), 389–339 | DOI

[103] R. R. Nigmatullin, “The realization of the generalized transfer equation in a medium with fractal geometry”, Phys. Stat. Sol. (b), 133:1 (1986), 425–430 | DOI

[104] A. A. Khamzin, R. R. Nigmatullin, I. I. Popov, “Mikroskopicheskaya model nedebaevskoi dielektricheskoi relaksatsii. Zakon Koula–Koula i ego obobschenie”, TMF, 173:2 (2012), 314–332 | DOI | DOI

[105] R. Balescu, “Anomalous transport in turbulent plasmas and continuous time random walks”, Phys. Rev. E, 51:5 (1995), 4807–4822 | DOI

[106] M. Tribeche, P. K. Shukla, “Charging of a dust particle in a plasma with a non extensive electron distribution function”, Phys. Plasmas, 18:10 (2011), 103702 | DOI

[107] V. E. Tarasov, “Electromagnetic field of fractal distribution of charged particles”, Phys. Plasmas, 12:8 (2005), 082106, 9 pp. | DOI

[108] A. S. Monin, “Uravneniya turbulentnoi diffuzii”, Dokl. AN SSSR. Ser. geogr. i geofiz., 105 (1955), 256 | MR | Zbl

[109] G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport”, Phys. Rep., 371:6 (2002), 461–580 | DOI | MR | Zbl

[110] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integraly i proizvodnye drobnogo poryadka i nekotorye ikh primeneniya, Nauka i tekhnika, Minsk, 1987 | MR | MR | Zbl

[111] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198, Academic Press, San Diego, CA, 1999 | MR

[112] V. E. Tarasov, “Fractional generalization of Liouville equations”, Chaos, 14:1 (2004), 123–127 | DOI

[113] V. E. Tarasov, “Fractional Liouville and BBGKI equations”, J. Phys.: Conf. Ser., 7:7 (2005), 117–133 | DOI

[114] V. E. Tarasov, “Fractional systems and fractional Bogoliubov hierarchy equations”, Phys. Rev. E, 71:1 (2005), 011102, 12 pp. | DOI | MR

[115] V. E. Tarasov, “Fractional statistical mechanics”, Chaos, 16:3 (2006), 033108, 7 pp. | DOI | MR | Zbl

[116] V. E. Tarasov, “Transport equations from Liouville equations for fractional systems”, Internat. J. Modern Phys. B, 20:3 (2006), 341–353 | DOI | MR | Zbl

[117] K. Cottrill-Shepherd, M. Naber, “Fractional differential forms”, J. Math. Phys., 42:5 (2001), 2203–2212 | DOI | MR | Zbl

[118] F. Mainardi, “Fractional calculus. Some basic problems in continuum and statistical mechanics”, Fractals and Fractional Calculus in Continuum Mechanics (Udine, September 23–27, 1996), CISM International Centre for Mechanical Sciences. Courses and Lectures, 378, eds. A. Carpinteri, F. Mainardi, Springer, Vienna, 1997, 291–348 | DOI | MR

[119] M. Caputo, F. Mainardi, “A new dissipation model based on memory mechanism”, Pure Appl. Geophys., 91:1 (1971), 134–147 | DOI

[120] P. Kostrobij, B. Markovych, O. Viznovych, M. Tokarchuk, “Generalized diffusion equation with fractional derivatives within Renyi statistics”, J. Math. Phys., 57:9 (2016), 093301, 8 pp. | DOI | MR | Zbl

[121] A. Compte, R. Metzler, “The generalized Cattaneo equation for the description of anomalous transport processes”, J. Phys. A: Math. Gen., 30:21 (1997), 7277–7289 | DOI | MR | Zbl

[122] I. I. Grygorchak, F O. Ivashchyshyn, M. V. Tokarchuk, N. T. Pokladok, O. V. Viznovych, “Modification of properties of GaSe($\beta$-cyclodexterin(FeSO$_4$)) Clathrat by synthesis in superposed electric and light-wave fields”, J. Appl. Phys., 121:18 (2017), 185501, 7 pp. | DOI

[123] T. D. Lee, “The strongly interacting quark-gluon plasma and future physics”, Nucl. Phys. A, 750:1 (2005), 1–8 | DOI

[124] M. Gyulassy, I. McLerran, “New forms of QCD matter discovered at RHIC”, Nucl. Phys. A, 750:1 (2005), 30–63 | DOI

[125] E. V. Shuryak, “What RHIC experiments and theory tell us about properties of quark-gluon plasma?”, Nucl. Phys. A, 750:1 (2005), 64–83 | DOI

[126] T. Hirano, N. Van Der Kolk, A. Bilandzic, “Hydrodynamics and flow”, The Physics of the Quark-Gluon Plasma. Introductory Lectures, Lecture Notes in Physics, 785, eds. S. Sarkar, H. Satz, B. Sinha, Springer, New York, 2010, 139–178 | DOI

[127] T. S. Biró, E. Molnár, “Non-extensive statistics, relativistic kinetic theory and fluid dynamics”, Eur. Phys. J. A, 48:11 (2012), 172, 11 pp. | DOI

[128] T. Osada, G. Wilk, “Nonextensive/dissipative correspondence in relativistic hydrodynamics”, Progr. Theor. Phys. Suppl., 174 (2008), 168–172 | DOI | Zbl

[129] T. Osada, G. Wilk, “Nonextensive hydrodynamics for relativistic heavy-ion collisions”, Phys. Rev. C., 77:4 (2008), 044903, 20 pp. | DOI

[130] T. Osada, G. Wilk, “Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics?”, Cent. Eur. J. Phys., 7:3 (2009), 432–443 | DOI

[131] T. Osada, “Relativistic hydrodynamical model in the presence of long-range correlations”, Phys. Rev. C., 81:2 (2010), 024907, 8 pp. | DOI

[132] A. Lavagno, “Relativistic nonextensive thermodynamics”, Phys. Lett. A, 301:1–2 (2002), 13–18 | DOI | MR | Zbl

[133] G. Gianpiero, A. Lavagno, D. Pigato, “Nonextensive statistical effects in the quark-gluon plasma formation at relativistic heavy-ion collisions energies”, Open Phys., 10:3 (2012), 594–601 | DOI

[134] A. Lavagno, D. Pigato, “Nonextensive statistical effects and strangeness production in hot and dense nuclear matter”, J. Phys. G: Nucl. Part., 39:12 (2012), 125106, 16 pp. | DOI

[135] A. Lavagno, D. Pigato, “Nonextensive nuclear liquid-gas phase transition”, Phys. A, 392:20 (2013), 5164–5171 | DOI | MR

[136] Kh. Umedzava, Kh. Matsumoto, M. Tatiki, Termopolevaya dinamika i kondensirovannye sostoyaniya, Mir, M., 1985

[137] L. Leplae, H. Umezawa, F. Mancini, “Derivation and application of the boson method in superconductivity”, Phys. Rep., 10:4 (1974), 151–272 | DOI

[138] Y. Takahashi, H. Umezawa, “Thermo field dynamics”, Collect. Phenom., 2:2 (1975), 55–80 ; reprinted in Internat. J. Modern Phys. B, 10:13–14 (1996), 1755–1805 | MR | DOI | MR | Zbl

[139] T. Arimitsu, H. Umezawa, “A general formulation of nonequlibrium thermo field dynamics”, Progr. Theor. Phys., 74:2 (1985), 429–432 | DOI | MR | Zbl

[140] T. Arimitsu, H. Umezawa, “Non-equilibrium thermo field dynamics”, Progr. Theor. Phys., 77:1 (1987), 32–52 | DOI

[141] T. Arimitsu, “A canonical formalism of dissipative quantum systems. Non-equilibrium thermofield dynamics”, Condens. Matter Phys., 4 (1994), 26–88 | DOI