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Exact and asymptotic properties of multistate random walks. (English) Zbl 0943.82573

Summary: A method is presented which allows one to obtain explicit analytical expressions (both exact and asymptotic) for many of the physically interesting quantities related to a multistate random walk (MRW). The exact results include the Laplace-Fourier-transformed probability distribution (continuous time) and generating function (discrete time), and closed evolution equations for the propagators related to each “internal” state of the walker. Analytical expressions for the scattering dynamical structure function and the frequency-dependent diffusion coefficient are given as illustrations. Asymptotic approximations to the single-state propagators are derived, allowing a detailed analysis of the longtime behavior and the calculation of asymptotic properties by single-state random walk standard methods. As an example, analytical expressions for the drift and diffusion coefficients are given.

MSC:

82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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[1] G. H. Weiss and R. J. Rubin,Adv. Chem. Phys. 32:363 (1983).
[2] E. W. Montroll and B. J. West, inFluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).
[3] J. W. Haus and K. W. Kehr,Phys. Rep. 150:263 (1987). · doi:10.1016/0370-1573(87)90005-6
[4] U. Landman and M. F. Shlesinger,Phys. Rev. B 16:3389 (1977);19:6207, 6220 (1979). · doi:10.1103/PhysRevB.16.3389
[5] J. B. T. M. Roerdink and K. E. Shuler,J. Stat. Phys. 40:205 (1985);41:581 (1985). · doi:10.1007/BF01010534
[6] A. Pesci,Phys. Lett. 112A:49 (1985).
[7] R. Bourret,Can. J. Phys. 38:665 (1960).
[8] M. O. Cáceres and H. S. Wio,Physica 142A:563 (1987).
[9] C. B. Briozzo, C. E. Budde, and M. O. Cáceres,Physica 159A:225 (1989).
[10] M. O. Cáceres and H. S. Wio,Z. Phys. B 54:175 (1984); H. S. Wio and M. O. Cáceres,Ann. Nucl. Energy 12:263 (1985). · doi:10.1007/BF01388069
[11] C. B. Briozzo, C. E. Budde, and M. O. Cáceres,Phys. Rev. A 39:6010 (1989). · doi:10.1103/PhysRevA.39.6010
[12] N. G. van Kampen,Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981). · Zbl 0511.60038
[13] M. O. Cáceres and C. E. Budde,Phys. Lett. 125A:369 (1987);Physica 153A:315 (1988).
[14] U. Landman, E. W. Montrol, and M. F. Shlesinger,Proc. Natl. Acad. Sci. USA 74:430 (1977); M. O. Cáceres,Phys. Rev. A 33:647 (1986). · doi:10.1073/pnas.74.2.430
[15] P. Hänggi and H. Thomas,Phys. Rep. 88:207 (1982). · doi:10.1016/0370-1573(82)90045-X
[16] K. Huffman and R. Kunze,Linear Algebra (Prentice-Hall, Englewood Cliffs, New Jersey, 1971).
[17] B. Davies,Integral Transforms and Their Applications (Springer-Verlag, New York, 1978). · Zbl 0381.44001
[18] W. Feller,An Introduction to Probability Theory and Its Applications (Wiley, New York, 1966). · Zbl 0138.10207
[19] U. W. Hochstrasser, inHandbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York).
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