Modeling of linear response for quantum nonextensive system on dynamic external disturbance

A. V. Kolesnichenko

Matematičeskoe modelirovanie, Tome 31 (2019) no. 12, pp. 97-118.

Résumé

In the framework of quantum statistical mechanics, based on the parametric nonadditive entropy of Tsallis, related to the density matrix, a dynamic theory of linear response of nonextensive quasi-equilibrium many-body systems to an external time-dependent perturbation is developed. In this paper, for the nonextensive quantum system proposed a modification of the Kubo theory developed in the framework of classical quantum mechanics. The construction of the microscopic theory of the linear reaction was carried out on the basis of the generalized canonical type of the density matrix, obtained by maximizing the Tsallis quantum entropy by averaging the observed values over the escort distribution. The generalized expressions for the admittance and the response function are presented, which describe the linear dependence of the system on a weak external mechanical action. The symmetry property for the relaxation function under time reversal and the Onsager reciprocity relation for generalized susceptibility are discussed. It is shown that these properties known in classical quantum statistics also remain valid for anomalous systems.

Détail

Keywords: quantum nonextensive statistics, generalized canonical distribution of the density matrix, linear reaction of the system.

Bibliographie

[1] J. Havrda, F. Charvat, “Quantiication method of classiication processes. Concept of structural $\alpha$-entropy”, Kybernetika, 3 (1967), 30–35 | MR | Zbl

[2] Z. Daroczy, “Generalized information functions”, Inf. Control., 16:1 (1970), 36–51 | MR | Zbl

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

[4] A. Renyi, “On measures of entropy and information”, Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, v. 1, University California Press, Berkeley, 1961, 547–561 | MR

[5] A. Renyi, Probability Theory, North-Holland Publ. Co, Amsterdam, 1970, 573 pp. | MR | Zbl

[6] E. M. F. Curado, C. Tsallis, “Generalized statistical mechanics: connection with thermodynamics”, J. Phys. A: Mathematical and General, 24:2 (1991), L69–72 | MR

[7] C. Beck, F. Schlogl, Thermodynamics of chaotic systems: an introduction, Cambridge University Press, Cambridge, 1993, 286 pp. | MR

[8] C. Tsallis, R. Mendes, A. Plastino, “The role of constraints within generalized nonextensive statistics”, Physica A, 261 (1998), 543–554

[9] C. Tsallis, Introduction to Nonextensive Statistical Mechanics. Approaching a Complex World, Springer, New York, 2009, 382 pp. | MR | Zbl

[10] A. Plastino, A. R. Plastino, “On the universality of thermodynamics' Legendre transform structure”, Phys. Lett. A, 226:5 (1997), 257–263 | MR | Zbl

[11] U. Tirnakli, D. F. Torres, “Exact and approximate results of non-extensive quantum statistics”, Eur. J. Phys. B, 14:4 (2000), 691–698

[12] E. K. Lenzi, R. S. Mendes, “Collisionless Boltzmann equation for systems obeying Tsallis distribution”, Eur. J. Phys. B, 21:3 (2001), 401–406

[13] S. Abe, “Heat and entropy in nonextensive thermodynamics: transmutation from Tsallis theory to Renyi-entropy-based theory”, Physica A: Statistical Mechanics and its Applications, 300:3 (2001), 417–423 | MR | Zbl

[14] R. G. Zaripov, Samoorganizatsiy i neobratimosty v neekstensivnykh sistemakh, Fen, Kazan, 2002, 251 pp.

[15] R. G. Zaripov, Printsipy neekstensivnoy statisticheskoy mekhaniki i geometrii mer besporyadka i poryadka, Izd. Kazanskogo Gostekhuniversiteta, Kazan, 2010, 404 pp.

[16] M. Gell-Mann, C. Tsallis, Nonextensive Entropy-Interdisciplinary Applications, Oxford University Press, eds. Eds, 2004, 440 pp. | MR | Zbl

[17] A. V. Kolesnichenko, “K postroeniyu neadditivnoy termodinamiki slozhnykh sistem na osnove statistiki Kurado-Tsallisa”, Preprinty IPM im. M.V. Keldysha, 2018, 025, 40 pp.

[18] A. V. Kolesnichenko, “K konstruirovaniyu termodinamiki neadditivnykh sred na osnove statistiki Tsallisa–Mendesa–Plastino”, Preprinty IPM im. M.V. Keldysha, 2018, 023, 28 pp.

[19] A. V. Kolesnichenko, “Dvukhparametricheskiy entropiynyy funktsional Sharma–Mittala kak osnova semeystva obobshchennykh termodinanik neekstensivnykh system”, Mathematica Montisnigri, 42 (2018), 74–101

[20] A. V. Kolesnichenko, Statisticheskay mekhanika i termodinamika Tsallisa neadditivnykh system. Vvedenie v teoriiu i prilozheniia, Sinergetika: ot proshlogo k budushchemu, 87, LENAND, M., 2019, 360 pp.

[21] A. R. Plastino, M. Casas, A. Plastino, “A nonextensive maximum entropy approach to a family of nonlinear reaction-diffusion equations”, Phys. A: Statistical Mechanics and its Applications, 280:3 (2000), 289–303 | MR

[22] B. M. Boghosian, “Navier-Stocks Equations for Generalized Thermostatistics”, Bras. J. Phys., 29:1 (1999), 91–107

[23] A. V. Kolesnichenko, B. N. Chetverushkin, “Kinetic derivation of a quasihydrodinamic system of equations on the base of nonextensive statistics”, Russian Journal of Numerical Analysis and Mathematical Modelling, 28 (2013), 547–576 | MR | Zbl

[24] T. D. Frank, A. Daffertshofer, “Multivariate nonlinear Fokker-Planck equations and generalized thermostatistics”, Phys. A: Statistical Mechanics and its Applications, 292:1 (2001), 392–410 | MR | Zbl

[25] N. Ito, C. Tsallis, “Specific heat of the harmonic oscillator within generalized equilibrium statistics”, Nuovo Cimento D, 11:6 (1989), 907–911

[26] P. H. Chavanis, L. Delfini, “Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model”, Eur. Phys. J. B, 69:3 (2009), 389–429

[27] A. V. Kolesnichenko, “Kriteriy termicheskoy ustoychivosti i zakon raspredeleniy chastits dlia samogravitiruiushchikh astro-fizicheskikh system v ramkakh statistiki Tsallisa”, Mathematica Montisnigri, 37 (2016), 45–75

[28] A. V. Kolesnichenko, “K razrabotke statisticheskoy termodinamiki i tekhniki fraktalnogo analiza dlya neekstensivnykh system na osnove entropii i razlichayushchey informatsii Ren'i”, Preprinty IPM im. M.V. Keldysha, 2018, 060, 44 pp.

[29] A. Esquivel, A. Lazarian, “Tsallis Statistics as a Tool for Studying Interstellar Turbulence”, Astrophys. J., 710:1 (2010), 125–132

[30] A. V. Kolesnichenko, M. Ya. Marov, “Modification of the jeans instability criterion for fractal-structure astrophysical objects in the framework of nonextensive statistics”, Solar System Research, 48:5 (2014), 354–365 | MR

[31] A. V. Kolesnichenko, “Power Distributions for Self-Gravitating Astrophysical Systems Based on Nonextensive Tsallis Kinetics”, Solar System Research, 51:2 (2017), 127–144

[32] A. V. Kolesnichenko, “Modifikatsiya v ramkakh statistiki Tsallisa kriteriev gravitatsionnoy neustoychivosti astrofisicheskikh diskov s fraktalnoy strukturoy fazovogo prostranstva”, Mathematica Montisnigri, 32 (2015), 93–118

[33] A. V. Kolesnichenko, “On construction of the entropy transport model based on the formalism of nonextensive statistics”, Math. Mod. Comp. Simul., 6:6 (2014), 587–597 | MR

[34] A. Wehrl, “General properties of entropy”, Reviews of Modern Physics, 50:2 (1978), 221–260 | MR

[35] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955, 325 pp. | MR | Zbl

[36] S. Abe, A. K. Rajagopal, “Validity of the Second Law in Nonextensive Quantum Thermodynamics”, Physical Review Letters, 91 (2003), 120601 | MR

[37] S. Abe, “Nonadditive generalization of the quantum Kullback–Leibler divergence for measuring the degree of purification”, Physical Review A, 68:3 (2003), 032302

[38] S. Abe, “Quantum q-divergence”, Physica A: Statistical Mechanics and its Applications, 344:3 (2004), 359–365 | MR

[39] S. Abe, “Geometric effect in nonequilibrium quantum thermodynamics”, Physica A: Statistical Mechanics and its Applications, 372:2 (2006), 387–392

[40] D. Zubarev, V. Morozov, G. Röpke, Statistical Mechanics of Nonequilibrium Processes, v. 1, Basic Concepts, Kinetic Theory, Akademie-Verlag, Berlin, 1996, 375 pp. | MR | Zbl

[41] R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems”, J. Phys. Soc. Jap., 12:6 (1957), 570–586 | MR

[42] R. Kubo, M. Yokota, S. Nakajima, “Statistical-Mechanical Theory of Irreversible Processes. II. Response to Thermal Disturbance”, J. Phys. Soc. Jap., 12:11 (1957), 1203–1211 | MR

[43] E. K. Lenzi, R. S. Mendes, “Collisionless Boltzmann equation for systems obeying Tsallis distribution”, Eur. J. Phys. B, 21:3 (2001), 401–406

[44] E. K. Lenzi, R. S. Mendes, A. K. Rajagopal, “Green functions based on Tsallis nonextensive statistical mechanics: normalized q-expectation value formulation”, Physica A: Statistical Mechanics and its Applications, 286:3 (2000), 503–517 | MR | Zbl

[45] E. K. Lenzi, R. S. Mendes, A. K. Rajagopal, “Quantum statistical mechanics for nonextensive systems”, Physical Review E (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics), 59:2 (1999), 1398–1407

[46] S. Abe, Y. Okamoto (eds.), Nonextensive Statistical Mechanicsand Its Applications, Chapter II, Lecture Notes in Physics, Springer Verlag, Berlin–New York, 2001 | MR

[47] S. Abe, “Axioms and uniqueness theorem for Tsallis entropy”, Physics Letters A, 271:1–2 (2000), 74–79 | MR | Zbl

[48] A. M. Gleason, “Measures on the closed subspaces of a Hilbert space”, Mathematics Journal (Indiana University), 6 (1957), 885–893 | MR | Zbl

[49] A. V. Kolesnivhenko, “K postroeniiu termodinamiki kvantovykh neekstensivnykh sistrm v ramkakh statistiki Tsallisa”, Preprinty IPM im. M.V. Keldysha, 2019, 016, 44 pp.

[50] E. T. Jaynes, “Information theory and statistical mechanics”, Statistical Physics, Brandeis Lectures, 3, 1963, 160 | MR

[51] A. V. Kolesnivhenko, “K vyvodu simmetrichnosti matritsy kineticheskikh koeffizientov Onzagera v ramkakh neekstensivnoy statisticheskoy mekhaniki Tsallisa”, Preprinty IPM im. M.V. Keldysha, 2019, 15, 24 pp.