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Analytically solvable model in fractional kinetic theory
Phys. Rev. E 71, 061104 – Published 15 June, 2005
DOI: https://doi.org/10.1103/PhysRevE.71.061104
Abstract
In this article we give a general prescription for incorporating memory effects in phase space kinetic equation, and consider in particular the generalized “fractional” relaxation time model equation. We solve this for small-signal charge carriers undergoing scattering, trapping, and detrapping in a time-of-flight experimental arrangement in two ways: (i) approximately via the Chapman-Enskog scheme for the weak gradient, hydrodynamic regime, from which the fractional form of Fick’s law and diffusion equation follow; and (ii) exactly, without any limitations on gradients. The latter yields complete and exact expressions in terms of generalized Mittag-Lefler functions for experimentally observable quantities. These expressions enable us to examine in detail the transition from the nonhydrodynamic stage to the hydrodynamic regime, and thereby establish the limits of validity of Fick’s law and the corresponding fractional diffusion equation.
Article Text
References (19)
- I. M. Sokolov, J. Klafter, and A. Blumen, Phys. Today, 55(11), 48(2002).
- H. Schiessel, Chr. Friedrich, and A. Blumen, in Applications of Fractional Calculus in Physics, edited by R. Hilfer (World Scientific, River Edge, NJ, 2000).
- R. Metzler and T. F. Nonnenmacher, Chem. Phys. 284, 67 (2002).
- R. Zallen, The Physics of Amorphous Solids (Wiley, New York, 1983).
- R. E. Robson, Aust. J. Phys. 28, 523 (1975).
- L. G. H. Huxley and R. W. Crompton, The Diffusion and Drift of Electrons in Gases (Wiley, New York, 1974); E. A. Mason and E. W. McDaniel, Transport Properties of Ions in Gases (Wiley, New York, 1988).
- K. Kumar, H. R. Skullerud, and R. E. Robson, Aust. J. Phys. 33, 343 (1980); K. Kumar, Phys. Rep. 112, 319 (1984); R. D. White, K. F. Ness, and R. E. Robson, Appl. Surf. Sci. 192, 26 (2002).
- E. M. Conwell, Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1967), Suppl. 9; Hot Electron Transport in Semiconductors, edited by L. Regianni (Springer-Verlag, Berlin, 1985); K. Seeger, Semiconductor Physics (Springer-Verlag, Berlin, 1989). The one-to-one correspondence between electron transport in semiconductors and gases has long been realized—see, for example, R. E. Robson, Phys. Rev. Lett. 31, 825 (1973).
- See, for example, K. Huang, Statistical Mechanics (Wiley, New York, 1963).
- T. H. Lovaas, H. R. Skullerud, O-H. Kristensen, and D. Linhjell, J. Phys. D 20, 1465 (1987).
- J. R. Haynes and W. Shockley, Phys. Rev. 81, 835 (1951).
- R. W. Crompton, Adv. At., Mol., Opt. Phys. 32, 97 (1994).
- R. E. Robson, B. Li, and R. D. White, J. Phys. B 33, 507 (2000).
- P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954).
- A. Erdélyi et al., Higher Transcendental Functions (McGraw-Hill, New York, 1955), Vol. III.
- H. Scher and E. W. Montroll, Phys. Rev. B 12, 2455 (1975).
- E. Barkai and R. L. Silbey, J. Phys. Chem. B 104, 3866 (2000). A different form of fractional kinetic equation, also with a Kramers (or Fokker-Planck) representation of the interaction term , has been proposed by R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000). The latter authors effectively include memory effects in the “flow” terms in the left-hand side of Eq. (7) of the present paper, as well as in . Dynamical trapping processes described by the left-hand side of the Boltzmann kinetic equation are considered in the review of G. M. Zaslavsky, ibid. 371, 461 (2002).
- B. D. Fried and S. D. Conte, The Plasma Dispersion Function (Academic Press, New York, 1961).
- F. Sigeneger and R. Winkler, Plasma Chem. Plasma Process. 17, 1 (1997); G. Petrov and R. Winkler, J. Phys. D 30, 53 (1997).