Open Access

Fractional statistical dynamics and fractional kinetics


Abstract

We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.

Key words: Configuration space, Caputo derivative, Vlasov-type kinetic equation, correlation functions, Poisson flow.


Full Text






Article Information

TitleFractional statistical dynamics and fractional kinetics
SourceMethods Funct. Anal. Topology, Vol. 22 (2016), no. 3, 197-209
MathSciNet   MR3554648
zbMATH 06742106
Milestones  Received 17/03/2016; Revised 03/04/2016
CopyrightThe Author(s) 2016 (CC BY-SA)

Authors Information

José Luís da Silva
CCM, University of Madeira, Campus da Penteada, 9020-105 Funchal, Portugal

Anatoly N. Kochubei
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

Yuri Kondratiev
Department of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany


Export article

Save to Mendeley



Citation Example

José Luís da Silva, Anatoly N. Kochubei, and Yuri Kondratiev, Fractional statistical dynamics and fractional kinetics, Methods Funct. Anal. Topology 22 (2016), no. 3, 197-209.


BibTex

@article {MFAT890,
    AUTHOR = {da Silva, José Luís and Kochubei, Anatoly N. and Kondratiev, Yuri},
     TITLE = {Fractional statistical dynamics and fractional kinetics},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {3},
     PAGES = {197-209},
      ISSN = {1029-3531},
  MRNUMBER = {MR3554648},
 ZBLNUMBER = {06742106},
       URL = {http://mfat.imath.kiev.ua/article/?id=890},
}


References

  1. Emilia Grigorova Bajlekova, Fractional evolution equations in Banach spaces, Eindhoven University of Technology, Eindhoven, 2001.  MathSciNet
  2. Emilia G. Bazhlekova, Subordination principle for fractional evolution equations, Fract. Calc. Appl. Anal. 3 (2000), no. 3, 213-230.  MathSciNet
  3. Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 30, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1961.  MathSciNet
  4. N. N. Bogoliubov, Problems of a dynamical theory in statistical physics, Studies in Statistical Mechanics, Vol. I, North-Holland, Amsterdam; Interscience, New York, 1962, pp. 1-118.  MathSciNet
  5. B. L. J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes-integrals, Compositio Math. 15 (1964), 239-341.  MathSciNet
  6. R. A. Carmona and S. A. Molchanov, Stationary parabolic Anderson model and intermittency, Probab. Theory Related Fields 102 (1995), no. 4, 433-453.  MathSciNet CrossRef
  7. Rene A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc. 108 (1994), no. 518, viii+125.  MathSciNet CrossRef
  8. José Luís Da Silva and Maria João Oliveira, Studies in fractional Poisson measures, International J. Modern Phys. Conf. Series 17 (2012), 122-129. CrossRef
  9. Dmitri Finkelshtein, Yuri Kondratiev, Yuri Kozitsky, and Oleksandr Kutoviy, The statistical dynamics of a spatial logistic model and the related kinetic equation, Math. Models Methods Appl. Sci. 25 (2015), no. 2, 343-370.  MathSciNet CrossRef
  10. Dmitri Finkelshtein, Yuri Kondratiev, and Oleksandr Kutoviy, Vlasov scaling for stochastic dynamics of continuous systems, J. Stat. Phys. 141 (2010), no. 1, 158-178.  MathSciNet CrossRef
  11. Dmitri Finkelshtein, Yuri Kondratiev, and Oleksandr Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal. 262 (2012), no. 3, 1274-1308.  MathSciNet CrossRef
  12. Dmitri Finkelshtein, Yuri Kondratiev, and Oleksandr Kutoviy, Statistical dynamics of continuous systems: perturbative and approximative approaches, Arab. J. Math. (Springer) 4 (2015), no. 4, 255-300.  MathSciNet
  13. Rudolf Gorenflo, Yuri Luchko, and Francesco Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal. 2 (1999), no. 4, 383-414.  MathSciNet
  14. H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math. (2011), Art. ID 298628, 51.  MathSciNet CrossRef
  15. Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.  MathSciNet
  16. A. N. Kochubei and Kondratiev. Y., Intermittency property for random point processes, in preparation, 2016.
  17. Yuri G. Kondratiev and Tobias Kuna, Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), no. 2, 201-233.  MathSciNet CrossRef
  18. Yu. G. Kondratiev and O. V. Kutoviy, On the metrical properties of the configuration space, Math. Nachr. 279 (2006), no. 7, 774-783.  MathSciNet CrossRef
  19. Yuri Kondratiev, Oleksandr Kutoviy, and Sergey Pirogov, Correlation functions and invariant measures in continuous contact model, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), no. 2, 231-258.  MathSciNet CrossRef
  20. Nick Laskin, Fractional Poisson process, Commun. Nonlinear Sci. Numer. Simul. 8 (2003), no. 3-4, 201-213.  MathSciNet CrossRef
  21. A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rational Mech. Anal. 59 (1975), no. 3, 219-239.  MathSciNet
  22. A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures, Arch. Rational Mech. Anal. 59 (1975), no. 3, 241-256.  MathSciNet
  23. F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics (Udine, 1996), CISM Courses and Lectures, vol. 378, Springer, Vienna, 1997, pp. 291-348.  MathSciNet CrossRef
  24. Francesco Mainardi, Rudolf Gorenflo, and Enrico Scalas, A fractional generalization of the Poisson processes, Vietnam J. Math. 32 (2004), no. Special Issue, 53-64.  MathSciNet
  25. Francesco Mainardi, Rudolf Gorenflo, and Alessandro Vivoli, Beyond the Poisson renewal process: a tutorial survey, J. Comput. Appl. Math. 205 (2007), no. 2, 725-735.  MathSciNet CrossRef
  26. Francesco Mainardi, Antonio Mura, and Gianni Pagnini, The $M$-Wright function in time-fractional diffusion processes: a tutorial survey, Int. J. Differ. Equ. (2010), Art. ID 104505, 29.  MathSciNet
  27. Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16 (2011), no. 59, 1600-1620.  MathSciNet CrossRef
  28. A. Mura, M. S. Taqqu, and F. Mainardi, Non-Markovian diffusion equations and processes: analysis and simulations, Phys. A 387 (2008), no. 21, 5033-5064.  MathSciNet CrossRef
  29. Maria Joao Oliveira and Rui Vilela Mendes, Fractional Boson gas and fractional Poisson measure in infinite dimensions, From particle systems to partial differential equations. II, Springer Proc. Math. Stat., vol. 129, Springer, Cham, 2015, pp. 293-312.  MathSciNet CrossRef
  30. Igor 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, vol. 198, Academic Press, Inc., San Diego, CA, 1999.  MathSciNet
  31. Jan Pruss, Evolutionary integral equations and applications, Modern Birkhauser Classics, Birkhauser/Springer Basel AG, Basel, 1993.  MathSciNet CrossRef
  32. O. N. Repin and A. I. Saichev, Fractional Poisson law, Radiophys. and Quantum Electronics 43 (2000), no. 9, 738-741 (2001).  MathSciNet CrossRef
  33. V. V. Uchaikin, D. O. Cahoy, and R. T. Sibatov, Fractional processes: from Poisson to branching one, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), no. 9, 2717-2725.  MathSciNet CrossRef


All Issues