Abstract
Based on the random-trap model and using the mean-field approximation, we derive an equation that allows the distribution of a functional of the trajectory of a particle making random walks over inhomogeneous-lattice site to be calculated. The derived equation is a generalization of the Feynman-Kac equation to an inhomogeneous medium. We also derive a backward equation in which not the final position of the particle but its position at the initial time is used as an independent variable. As an example of applying the derived equations, we consider the one-dimensional problem of calculating the first-passage time distribution. We show that the average first-passage times for homogeneous and inhomogeneous media with identical diffusion coefficients coincide, but the variance of the distribution for an inhomogeneous medium can be many times larger than that for a homogeneous one.
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References
A. H. Gandjbakhche and G. H. Weiss, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 61, 6958 (2000).
A. Bar-Haim and J. Klafter, J. Chem. Phys. 109, 5187 (1998).
N. Agmon, J. Chem. Phys. 81, 3644 (1984).
D. S. Grebenkov, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 76, 041139 (2007).
D. S. Grebenkov, Rev. Mod. Phys. 79, 1077 (2007).
G. Foltin, K. Oerding, Z. Racz, R. L. Workman, and R. P. K. Zia, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 50, R639 (1994).
M. Kac, Trans. Am. Math. Soc. 65, 1 (1949).
S. N. Majumdar, Curr. Sci. 89, 2076 (2005).
S. N. Majumdar and A. Comtet, Phys. Rev. Lett. 89, 060601 (2002).
S. Sabhapandit, S. N. Majumdar, and A. Comtet, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 73, 051102 (2006).
S. Carmi, L. Turgeman, and E. Barkai, arXiv:condmat/1004.0943.
B. Berkowitz, A. Cortis, M. Dentz, and H. Scher, Rev. Geophys. 44, RG2003 (2006).
V. P. Shkilev, JETP 108(2), 356 (2009).
V. P. Shkilev, JETP 109(5), 852 (2009).
M. Kac, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, California, United States, 1951 (Berkeley, 1951), p. 189.
L. Turgeman, S. Carmi, and E. Barkai, Phys. Rev. Lett. 103, 190201 (2009).
S. Carmi, L. Turgeman, and E. Barkai, J. Stat. Phys. 141, 1071 (2010).
M. Dentz, A. Cortis, H. Scher, and B. Berkowitz, Adv. Water Resour. 27, 155 (2004).
R. T. Sibatov and V. V. Uchaikin, arXiv:condmat//1008.3969.
M. S. Mommer and D. Lebiedz, SIAM J. Appl. Math. 70, 112 (2009).
H. Scher and M. Lax, Phys. Rev. B: Solid State 7, 4502 (1973).
M. Lax and H. Scher, Phys. Rev. Lett. 39, 751 (1977).
T. Odagaki and M. Lax, Phys. Rev. B: Condens. Matter 24, 5284 (1981).
V. M. Kenkre, Z. Kalay, and P. E. Parris, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 79, 011114 (2009).
I. M. Sokolov, S. B. Yuste, J. J. Ruiz-Lorenzo, and K. Lindenberg, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 79, 051113 (2009).
J. W. Haus and K. W. Kehr, Phys. Rep. 150, 264 (1987).
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Original Russian Text © V.P. Shkilev, 2012, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2012, Vol. 141, No. 1, pp. 195–204.
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Shkilev, V.P. Equations for the distributions of functionals of a random-walk trajectory in an inhomogeneous medium. J. Exp. Theor. Phys. 114, 172–181 (2012). https://doi.org/10.1134/S1063776111150106
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DOI: https://doi.org/10.1134/S1063776111150106