Area coverage of radial Lévy flights with periodic boundary conditions

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 87, Issue 4, 2013, Pages

  Vahabi, Mahsa ^a,b   Schulz, Johannes H P ^a   Shokri, Babak ^b   Metzler, Ralf ^c,d  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^b SHAHID BEHESHTI UNIVERSITY   (Iran)

^c UNIVERSITY OF POTSDAM   (Germany)

^d TAMPERE UNIVERSITY OF TECHNOLOGY   (Finland)

Author keywords

[No Author keywords available]

Indexed keywords

BOX-COUNTING ALGORITHM; BOX-COUNTING METHOD; COMPLEMENTARY METHODS; FIRST-PASSAGE-TIME DENSITY; FRACTIONAL ORDER; LONG TRAJECTORIES; MATHEMATICAL EXPRESSIONS; PERIODIC BOUNDARY CONDITIONS;

BOUNDARY CONDITIONS; PROBABILITY DENSITY FUNCTION; TRAJECTORIES;

FRACTAL DIMENSION;

EID: 84877752068     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.87.042136     Document Type: Article

References (48)

1. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).
2. B. D. Hughes, Random Walks and Random Environments, Vol. 1: Random Walks (Oxford University Press, Oxford, UK, 1995).
3. J.-P. Bouchaud and A. Georges, Phys. Rep. PRPLCM 0370-1573 10.1016/0370-1573(90)90099-N 195, 127 (1990).
4. R. Metzler and J. Klafter, Phys. Rep. PRPLCM 0370-1573 10.1016/S0370-1573(00)00070-3 339, 1 (2000);
5. R. Metzler and J. Klafter, J. Phys. A 1361-6447 10.1088/0305-4470/37/31/R01 37, R161 (2004).
6. P. Lévy, Calcul des probabilités (Gauthier-Villars, Paris, 1925);
7. Théorie de l'addition des variables aléatoires (Gauthier-Villars, Paris, 1954).
8. M. F. Shlesinger and J. Klafter, in On Growth and Form, edited by, H. E. Stanley, and, N. Ostrowsky, (Nijhoff, Amsterdam, 1986).
9. G. M. Viswanathan, Nature (London) NATUAS 0028-0836 10.1038/44831 401, 911 (1999);
10. G. M. Viswanathan, Nature (London) NATUAS 0028-0836 10.1038/381413a0 381, 413 (1996).
11. G. Ramos-Fernandez, J. L. Mateos, O. Miramontes, G. Cocho, H. Larralde, and B. Ayala-Orozco, Behav. Ecol. Sociobiol. 10.1007/s00265-003-0700-6 55, 223 (2004).
12. D. W. Sims, Nature (London) NATUAS 0028-0836 10.1038/nature06518 451, 1098 (2008);
13. D. W. Sims, Nature (London) NATUAS 0028-0836 10.1038/nature09116 465, 1066 (2010).
14. R. Nathan, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.0800375105 105, 19052 (2008).
15. F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra, and G. M. Viswanathan, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.88.097901 88, 097901 (2002).
16. M. A. Lomholt, T. Ambjörnsson, and R. Metzler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.260603 95, 260603 (2005).
17. D. Brockmann, L. Hufnagel, and T. Geisel, Nature (London) NATUAS 0028-0836 10.1038/nature04292 439, 462 (2006);
18. M. C. González, C. A. Hidalgo, and A.-L. Barabási, Nature (London) NATUAS 0028-0836 10.1038/nature06958 453, 779 (2008).
19. L. Hufnagel, D. Brockmann, and T. Geisel, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.0308344101 101, 15124 (2004).
20. A. V. Chechkin, R. Metzler, V. Yu. Gonchar, J. Klafter, and L. V. Tanatarov, J. Phys. A 1361-6447 10.1088/0305-4470/36/41/L01 36, L537 (2003).
21. T. Koren, M. A. Lomholt, A. V. Chechkin, J. Klafter, and R. Metzler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.99.160602 99, 160602 (2007).
22. B. Dybiec, E. Gudowska-Nowak, and P. Hänggi, Phys. Rev. E 1539-3755 10.1103/PhysRevE.73.046104 73, 046104 (2006).
23. S. Condamin, O. Bénichou, and M. Moreau, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.75.021111 75, 021111 (2007).
24. A. V. Chechkin, V. Y. Gonchar, J. Klafter, R. Metzler, and L. V. Tanatarov, J. Stat. Phys. JSTPBS 0022-4715 10.1023/B:JOSS.0000028067.63365.04 115, 1505 (2004);
25. A. V. Chechkin, J. Klafter, V. Yu. Gonchar, R. Metzler, and L. V. Tanatarov, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.67.010102 67, 010102 (R) (2003).
26. S. Jespersen, R. Metzler, and H. C. Fogedby, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.59.2736 59, 2736 (1999).
27. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Random Variables (Addison-Wesley, Reading, MA, 1954).
28. A. V. Chechkin, V. Yu. Gonchar, and M. Szydlowski, Phys. Plasmas PHPAEN 1070-664X 10.1063/1.1421617 9, 78 (2002).
29. In one dimension, the fractal dimension of a Brownian walk is one, however, its similarity dimension or mass dimension is two.
30. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, UK, 1990).
31. R. N. Mantegna and H. E. Stanley, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.73.2946 73, 2946 (1994).
32. I. Koponen, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.52.1197 52, 1197 (1995).
33. H. Nakao, Phys. Lett. A PYLAAG 0375-9601 10.1016/S0375-9601(00)00059-1 266, 282 (2000).
34. A. V. Chechkin and V. Y. Gonchar, Chaos, Solitons & Fractals 10.1016/S0960-0779(99)00168-X 11, 2379 (2000).
35. A. Zoia, A. Rosso, and M. Kardar, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.76.021116 76, 021116 (2007).
36. S. V. Buldyrev, S. Havlin, A. Ya. Kazakov, M. G. E. da Luz, E. P. Raposo, H. E. Stanley, and G. M. Viswanathan, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.64.041108 64, 041108 (2001).
37. N. Krepysheva, L. Di Pietro, and M.-C. Néel, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.73.021104 73, 021104 (2006).
38. B. Baeumer and M. M. Meerschaert, J. Comput. Appl. Math. JCAMDI 0377-0427 10.1016/j.cam.2009.10.027 233, 2438 (2010).
39. E. Barkai, Y. Garini, and R. Metzler, Phys. Today PHTOAD 0031-9228 10.1063/PT.3.1677 65 (8), 29 (2012).
40. S. Burov, R. Metzler, and E. Barkai, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.1003693107 107, 13228 (2010).
41. S. Burov, J.-H. Jeon, R. Metzler, and E. Barkai, Phys. Chem. Chem. Phys. PPCPFQ 1463-9076 10.1039/c0cp01879a 13, 1800 (2011).
42. W. Deng and E. Barkai, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79.011112 79, 011112 (2009).
43. J. H. Jeon, H. Martinez-Seara Monne, M. Javanainen, and R. Metzler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.109.188103 109, 188103 (2012).
44. Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.058101 101, 058101 (2008).
45. A. V. Weigel, Proc. Nat. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.1016325108 108, 6438 (2011).
46. J. H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sorensen, L. Oddershede, and R. Metzler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.106.048103 106, 048103 (2011).
47. Compare the interesting discussion of nonergodic behavior in the presence of power-law statistics with diverging moments in optical lattices: E. Lutz, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93.190602 93, 190602 (2004).
48. See the mathematical discussion in M. Magdziarz and A. Weron, Ann. Phys. APNYA6 0003-4916 10.1016/j.aop.2011.04.015 326, 2431 (2011).