Fractal behavior in trapping and reaction: A random walk study. (English) Zbl 0587.60062

We investigate the trapping of a random walker in fractal structures (Sierpinski gaskets) with randomly distributed traps. The survival probability is determined from the number of distinct sites visited in the trap-free fractals. We show that the short-time behavior and the long-time tails of the survival probability are governed by the spectral dimension \(\tilde d.\) We interpolate between these two limits by introducing a scaling law.
An extension of the theory, which includes a continuous-time random walk on fractals, is discussed as well as the case of direct trapping. The latter case is shown to be governed by the fractal dimension \(\bar d.\)


60G50 Sums of independent random variables; random walks
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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[1] E. W. Montroll and B. J. West, inFluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).
[2] G. H. Weiss and R. J. Rubin,Adv. Chem. Phys. 52:363 (1983).
[3] M. Koiwa and S. Ishioka,J. Stat. Phys. 30:477 (1983).
[4] H. B. Rosenstock,Phys. Rev. 187:1166 (1969);SIAM J. Appl. Math. 27:457 (1974).
[5] D. L. Huber, inLaser Spectroscopy of Solids, W. M. Yen and P. M. Selzer, eds. (Springer-Verlag, Berlin, 1981).
[6] J. Klafter and R. Silbey,J. Chem. Phys. 74:3510 (1981).
[7] G. Zumofen and A. Blumen,Chem. Phys. Lett. 88:63 (1982).
[8] P. M. Richards,J. Stat. Phys. 30:497 (1983).
[9] S. Redner and K. Kang,Phys. Rev. Lett. 51:1729 (1983).
[10] M. F. Shlesinger and E. W. Montroll,Proc. Natl. Acad. Sci. U.SA. 81:1280 (1984).
[11] B. Mandelbrot,The Fractal Geometry in Nature (W. H. Freeman, San Francisco, 1982). · Zbl 0504.28001
[12] S. Havlin and D. Ben-Avraham,J. Phys. A: Math. Gen. 15:L311 (1982).
[13] S. Alexander and R. Orbach,J. Phys. Lett. 43:625 (1982).
[14] R. Rammal and G. Toulouse,J. Phys. Lett. 44:13 (1983).
[15] T. Witten and L. M. Sander,Phys. Rev. Lett. 47:1400 (1981).
[16] S. Alexander, C. Laermans, R. Orbach, and H. M. Rosenberg,Phys. Rev. B 28:4615 (1983).
[17] D. Avnir, D. Farin, and P. Pfeifer,J. Chem. Phys. 79:3566 (1983).
[18] P. Evesque,J. Physique 44:1217 (1983).
[19] P. W. Klymko and R. Kopelman,J. Phys. Chem. 87:4565 (1983).
[20] G. H. Fredrickson and C. W. Frank,Macromolecules 16:1198 (1983).
[21] A. Blumen and G. Zumofen,J. Chem. Phys. 77:5127 (1982).
[22] G. H. Weiss,Proc. Natl. Acad. Sci. U.S.A. 77:4391 (1980). · Zbl 0438.60058
[23] P. G. de Gennes,C. R. Acad. Sci. Ser. II 296:881 (1983).
[24] R. Rammal, inCommon Trends in Particle and Condensed Matter Physics, Proceedings of the Les Houches Summer School Session XXXVIII, R. Stora and G. Grynberg, eds. (North-Holland, Amsterdam, in press).
[25] S. Alexander, inPercolation Structures and Processes, G. Deutcher, R. Zahlen, and J. Adler [Ann. Isr. Phys. Soc. 5:149 (1983)].
[26] A. Blumen, J. Klafter, and G. Zumofen,Phys. Rev. B 28:6112 (1983).
[27] I. M. Lifshitz,Adv. Phys. 13:483 (1964).
[28] B. Ya. Balagurov and V. G. Vaks,Zh. Exp. Toar. Fiz. 65:1939 (1973) [English transi.Sov. Phys. JETP 38:968 (1974)].
[29] P. Grassberger and I. Procaccia,J. Chem. Phys. 77:6281 (1982).
[30] R. F. Kayser and J. B. Hubbard,Phys. Rev. Lett. 51:79 (1983).
[31] A. A. Ovchinnikov and Ya. B. Zeldovich,Chem. Phys. 28:215 (1978).
[32] M. D. Donsker and S. R. S. Varadhan,Commun. Pure Appl. Math. 28:525 (1975);32:721 (1979). · Zbl 0333.60077
[33] C. M. Bender and S. A. Orszag,Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978). · Zbl 0417.34001
[34] J. Klafter, G. Zumofen, and A. Blumen,J. Phys. Lett. 45:L49 (1984).
[35] G. Zumofen, A. Blumen, and J. Klafter,J. Phys. A 17:L479 (1984). · Zbl 0587.60062
[36] B. Movaghar, G. W. Sauer, D. Würtz, and D. L. Huber,Solid State Commun. 39:1179(1981).
[37] A. Blumen,Nuovo Cimento 63B:50 (1981).
[38] M. Inokuti and F. Hirayama,J. Chem. Phys. 43:1978 (1965).
[39] C. Tsang and R. A. Street,Philos. Mag. B37:601 (1978).
[40] A. Blumen and J. Klafter,Philos. Mag. B47:L5 (1983).
[41] J. Klafter and A. Blumen,J. Chem. Phys. 80:875 (1984).
[42] E. W. Montroll and G. H. Weiss,J. Math. Phys. 6:167 (1965). · Zbl 1342.60067
[43] H. Scher and E. W. Montroll,Phys. Rev. B 12:2455 (1975).
[44] H. Scher and M. Lax,Phys. Rev. B 7:4491, 4502 (1973).
[45] J. Klafter and R. Silbey,Phys. Rev. Lett. 44:55 (1980).
[46] A. Blumen, J. Klafter, and G. Zumofen,Phys. Rev. B 27:3429 (1983).
[47] M. F. Shlesinger,J. Chem. Phys. 70:4813 (1979).
[48] G. Zumofen, J. Klafter, and A. Blumen,J. Chem. Phys. 79:5131 (1983).
[49] A. Blumen, G. Zumofen, and J. Klafter,J. Lumin, (in press).
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