Liouville and Bogoliubov equations with fractional derivatives

Modern Physics Letters B

Volumn 21, Issue 5, 2007, Pages 237-248

  Tarasov, Vasily E ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Bogoliubov equation; Fractional derivatives; Fractional kinetics; Liouville equation

Indexed keywords

EID: 33947378770     PISSN: 02179849     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217984907012700     Document Type: Article

References (62)

1. I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999).
2. K. B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New York, 1974).
3. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993).
4. G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002) 461-580.
5. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, Oxford, 2005).
6. A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics (Springer, New York, 1997).
7. R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000).
8. V. E. Tarasov, Continuous medium model for fractal media, Phys. Lett. A 336 (2005) 167-174;
9. Fractional Fokker-Planck equation for fractal media, Chaos 15 (2005) 023102;
10. Possible experimental test of continuous medium model for fractal media, Phys. Lett. A 341 (2005) 467-472;
11. Fractional hydrodynamic equations for fractal media, Ann. Phys. 318 (2005) 286-307;
12. Wave equation for fractal solid string, Mod. Phys. Lett. B 19 (2005) 721-728;
13. Dynamics of fractal solid, Int. J. Mod. Phys. B 19 (2005) 4103-4114;
14. Fractional Chapman-Kolmogorov equation, Mod. Phys. Lett. B 21(4) (2007).
15. G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Physica D 76 (1994) 110-122.
16. A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos 7 (1997) 753-764.
17. G. M. Zaslavsky and M. A. Edelman, Fractional kinetics: From pseudochaotic dynamics to Maxwell's demon, Physica D 193 (2004) 128-147.
18. R. R. Nigmatullin, Fractional kinetic equations and universal decoupling of a memory function in mesoscale region, Physica A 363 (2006) 282-298;
19. A. V. Chechkin, V. Yu. Gonchar and M. Szydlowsky, Fractional kinetics for relaxation and superdiffusion in magnetic field, Physics of Plasmas 9 (2002) 78-88; R. K. Saxena, A. M. Mathai and
20. H. J. Haubold, On fractional kinetic equations, Astrophysics and Space Science 282 (2002) 281-287.
21. B. A. Carreras, V. E. Lynch and G. M. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Physics of Plasmas 8 (2001) 5096-5103.
22. L. M. Zelenyi and A. V. Milovanov, Fractal topology and strange kinetics: From percolation theory to problems in cosmic electrodynamics, Physics Uspekhi 47 (2004) 749-788.
23. V. E. Tarasov, Electromagnetic field of fractal distribution of charged particles, Physics of Plasmas 12 (2005) 082106;
24. Multipole moments of fractal distribution of charges, Mod. Phys. Lett. B 19 (2005) 1107-1118;
25. Magnetohydrodynamics of fractal media, Physics of Plasmas 13 (2006) 052107;
26. Electromagnetic fields on fractals, Mod. Phys. Lett. A 21 (2006) 1587-1600.
27. V. E. Tarasov, Gravitational field of fractal distribution of particles, Celts. Mech. Dynam. Astron. 19 (2006) 1-15.
28. F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math. 118 (2000) 283-299.
29. F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals 7 (1996) 1461-1477.
30. V. E. Tarasov and G. M. Zaslavsky, Dynamics with low-level fractionality, Physica A 368 (2006) 399-415.
31. V. E. Tarasov, Fractional generalization of Liouville Equation, Chaos 14 (2004) 123-127;
32. Fractional systems and fractional Bogoliubov hierarchy equations, Phys. Rev. E 71 (2005) 011102;
33. Fractional Liouville and BBGKI equations, J. Phys. Conf. Ser. 7 (2005) 17-33;
34. Transport equations from Liouville equations for fractional systems, Int. J. Mod. Phys. B 20 (2006) 341-354;
35. Fokker-Planck equation for fractional systems, Int. J. Mod. Phys. B 21 (2007) accepted.
36. V. E. Tarasov, Fractional generalization of gradient and Hamiltonian systems, J. Phys. A 38 (2005) 5929-5943;
37. Fractional generalization of gradient systems, Lett. Math. Phys. 73 (2005) 49-58;
38. Fractional variations for dynamical systems: Hamilton and Lagrange approaches, J. Phys. A 39 (2006) 8409-8425.
39. N. Laskin and G. M. Zaslavsky, Nonlinear fractional dynamics on a lattice with longrange interactions, Physica A 368 (2006) 38-54.
40. V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos 16 (2006) 023110;
41. Fractional dynamics of systems with long-range interaction, Commun. Nonlin. Sci. Numer. Simul. 11 (2006) 885-898.
42. N. Korabel, G. M. Zaslavsky and V. E. Tarasov, Coupled oscillators with power-law interaction and their fractional dynamics analogues, Commun. Nonlin. Sci. Numer. Simul. 11 (2006) accepted (math-ph/0603074).
43. K. Cottrill-Shepherd and M. Naber, Fractional differential forms, J. Math. Phys. 42 (2001) 2203-2212;
44. F. B. Adda, The differentiability in the fractional calculus, Nonlinear Analysis 47 (2001) 5423-5428.
45. M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and Appl. Geophys. 91 (1971) 134-147.
46. N. N. Bogoliubov, Kinetic equations, Zh. Exper. Tear. Fiz. 16 (1946) 691;
47. J. Phys. USSR 10 (1946) 265.
48. N. N. Bogoliubov, in Studies in Statistical Mechanics, Vol. 1, eds. J. de Boer and G. E. Uhlenbeck (North-Holland, Amsterdam, 1962), p. 1.
49. K. P. Gurov, Foundation of Kinetic Theory: Method of N. N. Bogoliubov (Nauka, Moscow, 1966) in Russian.
50. D. Ya. Petrina, V. I. Gerasimenko and P. V. Malishev, Mathematical Basis of Classical Statistical Mechanics (Naukova dumka, Kiev, 1985) in Russian.
51. N. N. Bogoliubov and N. N. Bogoliubov, Jr., Introduction to Quantum Statistical Mechanics (World Scientific, Singapore, 1982), Sec. 1.4.
52. A. A. Vlasov, Vibrating properties of electronic gas, Zh. Exper. Tear. Fix, 8 (1938) 291;
53. On the kinetic theory of an assembly of particles with collective interaction, J. Phys. USSR 9 (1945) 25.
54. A. A. Vlasov, Many-particle Theory and its Application to Plasma (Gordon and Breach, New York, 1961).
55. G. Ecker, Theory of Fully Ionized Plasmas (Academic Press, New York, 1972).
56. N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, New York, 1973).
57. V. E. Tarasov, Stationary solution of Liouville equation for non-Hamiltonian systems, Ann. Phys. 316 (2005) 393-413;
58. Classical canonical distribution for dissipative systems, Mod. Phys. Lett. B 17 (2003) 1219-1226.
59. A. Isihara, Statistical Physics (Academic Press, New York, London, 1971), Appendix IV, Sec. 7.5.
60. P. Resibois and M. De Leener, Classical Kinetic Theory of Fluids (John Wiley and Sons, New York, London, 1977), Sec. IX.4.
61. D. Forster, Hydrodynamics Fluctuations, Broken Symmetry, and Correlation Functions (W. E. Benjamin Inc., London, Amsterdam, 1975), Sec. 6.4.
62. V. Feller, An Introduction to Probability Theory and its Applications, Vol. 2 (Wiley, New York, 1971).