Abstract
The linear Boltzmann equation (LBE) approach is generalized to describe fractional superdiffusive transport of the Lévy walk type in external force fields. The time distribution between scattering events is assumed to have a finite mean value and infinite variance. It is completely characterized by the two scattering rates, one fractional and a normal one, which defines also the mean scattering rate. We formulate a general fractional LBE approach and exemplify it with a particularly simple case of the Bohm and Gross scattering integral leading to a fractional generalization of the Bhatnagar, Gross and Krook (BGK) kinetic equation. Here, at each scattering event the particle velocity is completely randomized and takes a value from equilibrium Maxwell distribution at a given fixed temperature. We show that the retardation effects are indispensable even in the limit of infinite mean scattering rate and argue that this novel fractional kinetic equation provides a viable alternative to the fractional Kramers–Fokker–Planck (KFP) equation by Barkai and Silbey and its generalization by Friedrich et al. based on the picture of divergent mean time between scattering events. The case of divergent mean time is also discussed at length and compared with the earlier results obtained within the fractional KFP. Also a phenomenological fractional BGK equation without retardation effects is proposed in the limit of infinite scattering rates. It cannot be, however, rigorously derived from a scattering model, being rather clever postulated. It this respect, this retardationless equation is similar to the fractional KFP by Barkai and Silbey. However, it corresponds to the opposite, much more physical limit and, therefore, also presents a viable alternative.
Similar content being viewed by others
References
M.R. Hoare, Adv. Chem. Phys. 20, 135 (1971)
L.P. Pitaevskii, E.M. Lifshitz, in Physical kinetics, Landau and Lifshitz course of theoretical physics (Pergamon Press, New York, 1981), Vol. 10
H. Risken, Fokker–Planck equation, methods of solution and applications, 2nd edn. (Springer, Berlin, 1989)
Yu.L. Klimontovich, Phys. Usp. 37, 737 (1994)
N.G. Van Kampen, Stochastic processes in physics and chemistry, 2nd edn. (North-Holland, Amsterdam, 1997)
D. Bohm, E.P. Gross, Phys. Rev. 75, 1864 (1949)
E. Barkai, V. Fleurov, Chem. Phys. 212, 69 (1996)
E. Barkai, J. Klafter, in Chaos, kinetics and nonlinear dynamics in fluids and plasmas, edited by S. Benkadda, G.M. Zaslavsky (Springer, Berlin, 1997)
E. Barkai, V.N. Fleurov, Phys. Rev. E 56, 6355 (1997)
P.L. Bhatnagar, E.P. Gross, M. Krook, Phys. Rev. 94, 511 (1954)
R. Zwanzig, Nonequilibrium statistical mechanics (Oxford University Press, Oxford, 2001)
S. Succi, I.V. Karlin, H. Chen, Rev. Mod. Phys. 74, 1203 (2002)
V.M. Kenkre, E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 9, 45 (1973)
M.F. Shlesinger, J. Stat. Phys. 10, 421 (1974)
H. Scher, E.M. Montroll, Phys. Rev. B 12, 2455 (1975)
J.-P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)
B.D. Hughes, in Random walks and random environments (Clarendon Press, Oxford, 1995), Vols. 1–2
R. Balescu, Statistical dynamics: matter out of equilibrium (Imperial College Press, London, 1997)
D. Ben-Avraham, Sh. Havlin, Diffusion and reactions in fractals and disordered systems (Cambridge University Press, Cambridge, 2000)
T. Geisel, A. Zacherl, G. Radons, Z. Phys. B 71, 117 (1988)
G. Zumofen, J. Klafter, Phys. Rev. E 47, 851 (1993)
G. Zumofen, J. Klafter, Physica D 69, 436 (1993)
B.J. West, P. Grigolini, R. Metzler, T.F. Nonnenmacher, Phys. Rev. E 55, 99 (1997)
R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
I. Goychuk, Phys. Rev. E 86, 021113 (2012)
E. Barkai, R.J. Silbey, J. Chem. Phys. B 104, 3866 (2000)
R. Friedrich, F. Jenko, A. Baule, S. Eule, Phys. Rev. Lett. 96, 230601 (2006)
R. Friedrich, F. Jenko, A. Baule, S. Eule, Phys. Rev. E 74, 041103 (2006)
R. Metzler, J. Klafter, Phys. Rev. E 61, 6308 (2000)
D.E. Cox, Renewal theory (Methuen, London, 1962)
C. Godreche, J.M. Luck, J. Stat. Phys. 104, 489 (2001)
P. Allegrini, G. Aquino, P. Grigolini, L. Palatella, A. Rosa, B.J. West, Phys. Rev. E 71, 066109 (2005)
G. Margolin, E. Barkai, J. Chem. Phys. 121, 1566 (2004)
D. Froemberg, E. Barkai, Eur. Phys. J. B 86, 331 (2013)
J.K.E. Tunaley, Phys. Rev. Lett. 33, 1037 (1974)
I. Goychuk, Commun. Theor. Phys. 62, 497 (2014)
I. Goychuk, P. Hänggi, Phys. Rev. Lett. 91, 070601 (2003)
I. Goychuk, Phys. Rev. E 70, 016109 (2004)
R. Gorenflo, F. Mainardi, in Fractals and fractional calculus in continuum mechanics, edited by A. Carpinteri, F. Mainardi (Springer, Wien, 1997), pp. 223–276
I.M. Sokolov, J. Klafter, Chaos 15, 026103 (2005)
A.V. Chechkin, V.Yu. Gonchar, R. Gorenflo, N. Korabel, I.M. Sokolov, Phys. Rev. E 78, 021111 (2008)
A. Papoulis, in Probability, random variables, and stochastic processes, 3rd edn. (McGraw-Hill Book Company, New York, 1991), pp. 430–432
F. Mainardi, P. Pironi, Extr. Math. 11, 140 (1996)
E. Lutz, Phys. Rev. Lett. 93, 190602 (2004)
P. Siegle, I. Goychuk, P. Hänggi, Europhys. Lett. 93, 20002 (2011)
I. Goychuk, Adv. Chem. Phys. 150, 187 (2012)
H. Stehfest, Commun. ACM 13, 47 (1970) [H. Stehfest,Commun. ACM 13, 624 (1970) (Erratum)]
I.M. Sokolov, R. Metzler, Phys. Rev. E 67, 010101(R) (2003)
S. Eule, R. Friedrich, F. Jenko, D. Kleinhans, J. Phys. Chem. B 111, 11474 (2007)
W.T. Coffey, Y.P. Kalmykov, The Langevin equation, with applications to stochastic problems in physics, chemistry and electrical engineering, 3rd edn. (World Scientific, New Jersey, 2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contribution to the Topical Issue “Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook”, edited by Ryszard Kutner and Jaume Masoliver.
Rights and permissions
About this article
Cite this article
Goychuk, I. Fractional Bhatnagar–Gross–Krook kinetic equation. Eur. Phys. J. B 90, 208 (2017). https://doi.org/10.1140/epjb/e2017-80297-x
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2017-80297-x