Geodesic
Complementary Densities of Lévy Walks: Typical and Rare Fluctuations
A. Rebenshtok1 ; S. Denisov2, 3, 4 ; P. Hänggi2 ; E. Barkai1
1 Department of Physics, Institute of Nanotechnology and Advanced Materials Bar-Ilan University, Ramat-Gan, 52900, Israel
2 Institute of Physics, University of Augsburg, Universitätsstrasse 1 D-86135, Augsburg Germany
3 Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine
4 Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 76-106.

Voir la notice de l'article provenant de la source EDP Sciences

Strong anomalous diffusion is a recurring phenomenon in many fields, ranging from the spreading of cold atoms in optical lattices to transport processes in living cells. For such processes the scaling of the moments follows 〈|x(t)|q〉 ∼ tqν(q) and is characterized by a bi-linear spectrum of the scaling exponents, qν(q). Here we analyze Lévy walks, with power law distributed times of flight ψ(τ) ∼ τ−(1+α), demonstrating sharp bi-linear scaling. Previously we showed that for α > 1 the asymptotic behavior is characterized by two complementary densities corresponding to the bi-scaling of the moments of x(t). The first density is the expected generalized central limit theorem which is responsible for the low-order moments 0 q α. The second one, a non-normalizable density (also called infinite density) is formed by rare fluctuations and determines the time evolution of higher-order moments. Here we use the Faà di Bruno formalism to derive the moments of sub-ballistic super-diffusive Lévy walks and then apply the Mellin transform technique to derive exact expressions for their infinite densities. We find a uniform approximation for the density of particles using Lévy distribution for typical fluctuations and the infinite density for the rare ones. For ballistic Lévy walks 0 α 1 we obtain mono-scaling behavior which is quantified.
DOI : 10.1051/mmnp/201611306

A. Rebenshtok 1 ; S. Denisov 2, 3, 4 ; P. Hänggi 2 ; E. Barkai 1

1 Department of Physics, Institute of Nanotechnology and Advanced Materials Bar-Ilan University, Ramat-Gan, 52900, Israel
2 Institute of Physics, University of Augsburg, Universitätsstrasse 1 D-86135, Augsburg Germany
3 Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine
4 Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia 
@article{MMNP_2016_11_3_a5,
     author = {A. Rebenshtok and S. Denisov and P. H\"anggi and E. Barkai},
     title = {Complementary {Densities} of {L\'evy} {Walks:} {Typical} and {Rare} {Fluctuations}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {76--106},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {2016},
     doi = {10.1051/mmnp/201611306},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611306/}
}
TY  - JOUR
AU  - A. Rebenshtok
AU  - S. Denisov
AU  - P. Hänggi
AU  - E. Barkai
TI  - Complementary Densities of Lévy Walks: Typical and Rare Fluctuations
JO  - Mathematical modelling of natural phenomena
PY  - 2016
SP  - 76
EP  - 106
VL  - 11
IS  - 3
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611306/
DO  - 10.1051/mmnp/201611306
LA  - en
ID  - MMNP_2016_11_3_a5
ER  - 
%0 Journal Article
%A A. Rebenshtok
%A S. Denisov
%A P. Hänggi
%A E. Barkai
%T Complementary Densities of Lévy Walks: Typical and Rare Fluctuations
%J Mathematical modelling of natural phenomena
%D 2016
%P 76-106
%V 11
%N 3
%I mathdoc
%U https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611306/
%R 10.1051/mmnp/201611306
%G en
%F MMNP_2016_11_3_a5
A. Rebenshtok; S. Denisov; P. Hänggi; E. Barkai. Complementary Densities of Lévy Walks: Typical and Rare Fluctuations. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 76-106. doi : 10.1051/mmnp/201611306. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611306/

[1] T. Akimoto 2012 164101

[2] T. Akimoto, T. Miyagichi 2014 515 530

[3] K. H. Andersen, P. Castiglione, A. Mazzino, A. Vulpiani 2000 447 452

[4] G. B. Arfken, H. J. Weber. Mathematical methods for physicists, Academic Press (1995).

[5] B. Baeumer, M.M. Meerschaert, J. Mortensen. Space time fractional derivative operators. Proc. Amer. Math. Soc. (ISSN: 0002-9939) 133 (8) (2005) 2273-2282.

[6] E. Barkai, E. Aghion, D. A. Kessler Phys. Rev. X. 2014 021036

[7] E. Barkai, J. Klafter. Anomalous diffusion in the strong scattering limit: a Lévy walk approach Proceedings of a workshop held in Carry-Le-Rouet, (1997) S. Benkadda and G. M. Zaslavsky Editors Springer (Berlin).

[8] P. Becker-Kern, M. Meerschaert, H. Scheffler Ann. Probab. 2004 730 756

[9] G. Bel, E. Barkai Phys. Rev. Lett. 2005

[10] J. P. Bouchaud, A. Georges Phys. Rep. 1998 127 293

[11] R. Burioni, L. Caniparoli, A. Vezzani Phys. Rev. E. 2010

[12] R. Burioni, G. Gradenigo, A. Sarracino, A. Vezzani, A. Vulpiani J. Stat. Mech. Theory and Experiment 2013

[13] G. Campagnola, K. Nepal, B. W. Schroder, O. B. Peersen, D. Krapf Scientific Reports 2015

[14] B. A. Carreras, V. E. Lynch, D. E. Newman, G. M. Zaslavsky Phys. Rev. E. 1999 4770

[15] P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, A. Vulpiani Physica D 1999 75 93

[16] Clearly, the strip of definition can be extended in some cases. For example, if B(v) is a Gaussian then M(1 / 2) is finite. The choice 1 ≤ Re(q) stems from normalization.

[17] L. Comtet. Advanced Combinatorics, D. Reidel Publishing Company (1974).

[18] P. De Anna, T. Le Borgne, M. Dentz, A. M. Tartakovsky, D. Bolster, P. Davy Phys. Rev. Lett. 2013

[19] M. Dentz, T. Le Borgne, D. R. Lester, F. P. J. De Barros Phys. Rev. E. 2015

[20] A. Dhar, K. Saito, B. Derrida Phys. Rev. E. 2013 010103

[21] S. Fedotov Phys. Rev. E. 2016 020101

[22] W. Feller. An introduction to probability theory and its applications 2, John Wiley and Sons, New York (1971).

[23] D. Froemberg, M. Schmiedeberg, E. Barkai, V. Zaburdaev Phys. Rev. E. 2015

[24] N. Gal, D. Weihs Phys. Rev. E. 2010

[25] C. Godréche, J. M. Luck J. Stat. Phys. 2001 489 524

[26] N. Hazut, S. Medalion, D. A. Kessler, E. Barkai Phys. Rev. E. 2015

[27] A. Jurlewicz, P. Kern, M. Meerschaert, H.-P. Scheffler Comput. Math. Appl. 2012 3021 3036

[28] D.A. Kessler, E. Barkai Phys. Rev. Lett. 2010

[29] J. Klafter, A. Blumen, M. F. Shlesinger Phys. Rev. A. 1987 3081 3085

[30] N. Korabel, E. Barkai Phys. Rev. Lett. 2000

[31] D. Krapf, G. Campagnola, K. Nepal, O. B. Peersen. Strange kinetics of bulk mediated diffusion in living cells. arXiv:1601.04198 [cond-mat.stat-mech] (2016).

[32] P. Lévy. Théorie de l’addition des variables aléatoires. (1937) Gauthiers-Villars, Paris.

[33] E. Lukacs American Mathematical Monthly 1955 340 348

[34] M Magdziarz, Hp Scheffler, P Straka, P Zebrowski Stochastic Processes and their Applications 2015 4021 4038

[35] B. B. Mandelbrot, J. W. Van Ness SIAM Rev. 1968 422 437

[36] G. Margolin, V. Protasenko, M. Kuno, E. Barkai J. of Physical Chemistry B 2006 19053

[37] R. Metzler, J. Klafter Physics Reports 2000 1 77

[38] Ed. A. D. Poularikas. The Transform and Applications Handbook, CRC Press inc (1995).

[39] A. Rebenshtok, E. Barkai Phys. Rev. Lett. 2007 210601

[40] A. Rebenshtok, E. Barkai J. Stat. Phys. 565 586 2008

[41] A. Rebenshtok, S. Denisov, P. Hänggi, E. Barkai Phys. Rev. Lett. 2014 110601

[42] A. Rebenshtok, S. Denisov, P. Hänggi, E. Barkai 2014 062135

[43] A. Rebenshtok, S. Denisov, P. Hänggi, E. Barkai. Reply to the comment on "Non-Normalizable Densities in Strong Anomalous Diffusion: Beyond the Central Limit Theorem". arXiv:1502.01749 [cond-mat.stat-mech] (2015).

[44] D. P. Sanders, H. Larralde Phys. Rev. E. 2006

[45] J. H. P. Schulz, E. Barkai Fluctuations around equilibrium laws in ergodic continuous-time random walks Phys. Rev. E. 2015 062129

[46] M. F. Shlesinger, B. J. West, J. Klafter Phys. Rev. Lett. 1987 1100 1103

[47] F. D. Stefani, J. P. Hoogenboom, E. Barkai Physics Today 2009 34 39

[48] P. Straka, B. Henry Stochast. Process. Appl. (ISSN: 03044149) 2011 324 336

[49] H. Touchette Physics Reports 2009 1 69

[50] A. Vulpiani, F. Cecconi, M. Cencini, A. Puglisi, D. Vergni (Eds). Large deviations in physics: the legacy of the law of large numbers. Lecture notes in physics (2014).

[51] G. H. Weiss. Aspects and Applications of the Random Walk, North-Holland, Amsterdam (1994) .

[52] V. Zaburdaev, S. Denisov, P. Hänggi Phys. Rev. Lett. 170604 2013

[53] V. Zaburdaev, S. Denisov, P. Hänggi Phys. Rev. Lett. 2011

[54] V. Zaburdaev, S. Denisov, J. Klafter Rev. Mod. Phys. 2015

[55] G. Zumofen, J. Klafter Phys. Rev. E. 1993 851 863

Cité par Sources :