Sub-diffusive scaling with power-law trapping times

doi:10.1088/1674-1056/23/7/070514

Abstract Thermally driven diffusive motion of a particle underlies many physical and biological processes. In the presence of traps and obstacles, the spread of the particle is substantially impeded, leading to subdiffusive scaling at long times. The statistical mechanical treatment of diffusion in a disordered environment is often quite involved. In this short review, we present a simple and unified view of the many quantitative results on anomalous diffusion in the literature, including the scaling of the diffusion front and the mean first-passage time. Various analytic calculations and physical arguments are examined to highlight the role of dimensionality, energy landscape, and rare events in affecting the particle trajectory statistics. The general understanding that emerges will aid the interpretation of relevant experimental and simulation results.

Keywords: anomalous diffusion subdiffusive scaling random walk in a disordered medium glass transition

Received: 08 April 2014
Revised: 29 May 2014
Accepted manuscript online:

PACS:	05.40.-a	(Fluctuation phenomena, random processes, noise, and Brownian motion)
	05.40.Fb	(Random walks and Levy flights)
	66.10.C-	(Diffusion and thermal diffusion)
	87.16.dp	(Transport, including channels, pores, and lateral diffusion)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11175013) and the Research Grants Council of the Hong Kong Special Administrative Region, China (Grant No. N HKBU 213/10).

Corresponding Authors: Tang Lei-Han
E-mail: lhtang@csrc.ac.cn

About author: 05.40.-a; 05.40.Fb; 66.10.C-; 87.16.dp

Cite this article:

Luo Liang (罗亮), Tang Lei-Han (汤雷翰) Sub-diffusive scaling with power-law trapping times 2014 Chin. Phys. B 23 070514

[1]	Bronstein I, Israel Y, Kepten E, Mai S, Shav-Tal Y, Barkai E and Garini Y 2009 Phys. Rev. Lett. 103 018102
[2]	Jeon J H, Tejedor V, Burov S, Barkai E, Selhuber-Unkel C, Berg- Srensen K, Oddershede L and Metzler R 2011 Phys. Rev. Lett. 106 048103
[3]	Weigel A V, Simon B, Tamkun A A and Krapf D 2011 Proc. Natl. Acad. Sci. USA 108 6438
[4]	Barkai E, Garini Y and Metzler R 2012 Phys. Today 65 29
[5]	Bénichou O, Chevalier C, Klafter J Meyer B and Voituriez R 2010 Nature Chem. 2 472
[6]	Berry H and Chaté H 2014 Phys. Rev. E 89 022708
[7]	See e.g. Saxton M J 2001 Biophys. J. 81 2226
[8]	Lepzelter D and Zaman M 2012 J. Chem. Phys. 137 175102 and references therein
[9]	Haus J W and Kehr K W 1987 Phys. Rep. 150 263
[10]	Bouchaud J P and Georges A 1990 Phys. Rep. 195 127
[11]	Iglói F and Monthus C 2005 Phys. Rep. 412 277
[12]	Redner S 2001 A Guide to First-Passage Processes (Cambridge University Press, Cambridge)
[13]	Montroll E W and Weiss G H 1965 J. Math. Phys. 6 167
[14]	Scher H and Montroll E W 1975 Phys. Rev. B 12 2455
[15]	Saichev A I and Zaslavsky G M 1997 Chaos 7 753
[16]	Barkai E 2001 Phys. Rev. E 63 046118
[17]	Meerschaert M M and Scheffler H P 2004 J. Appl. Probab. 41 623
[18]	Sokolov I M and Klafter J 2005 Chaos 15 026103
[19]	Yuste S B and Lindenberg K 2005 Phys. Rev. E 72 061103
[20]	Bouchaud J P 1992 J. Phys. I France 2 1705
[21]	Bel G and Barkai E 2005 Phys. Rev. Lett. 94 240602
[22]	He Y, Burov S, Metzler R and Barkai E 2008 Phys. Rev. Lett. 101 058101
[23]	Monthus C 2003 Phys. Rev. E 68 036114
[24]	Machta J 1981 Phys. Rev. B 24 5260
[25]	Machta J 1985 J. Phys. A 18 L531
[26]	Bertin E M and Bouchaud J P 2003 Phys. Rev. E 67 026128
[27]	Burov S and Barkai E 2011 Phys. Rev. Lett. 106 140602
[28]	Burov S and Barkai E 2012 Phys. Rev. E 86 041137
[29]	Sinaï Y G 1982 Theor. Probab. Appl. 27 247
[30]	Marinari E, Parisi G and Ruelle D and Windey P 1983 Phys. Rev. Lett. 50 1223
[31]	Marinari E, Parisi G and Ruelle D andWindey P 1983 Commun. Math. Phys. 89 1
[32]	Monthus C and Garel T 2010 Phys. Rev. E 81 011138
[33]	Fisher D S, Le Doussal P and Monthus C 1998 Phys. Rev. Lett. 80 3539
[34]	Le Doussal P, Monthus C and Fisher D S 1999 Phys. Rev. E 59 4795
[35]	Monthus C and Le Doussal P 2004 Physica A 334 78
[36]	Monthus C 2003 Phys. Rev. E 67 046109.
[37]	Carpentier D and Le Doussal P 2001 Phys. Rev. E 63 026110
[38]	Korshunov S E and Nattermann T 1996 Phys. Rev. B 53 2746
[39]	Tang L H 1996 Phys. Rev. B 54 3350
[40]	Derrida B and Spohn H 1988 J. Stat. Phys. 51 817
[41]	Derrida B 1981 Phys. Rev. B 24 2613
[42]	Derrida B and Toulouse G 1985 J. Phys. Lett. 46 223
[43]	Castillo H E and Le Doussal P 2001 Phys. Rev. Lett. 86 4859
[44]	Gorokhov D A and Blatter G 1998 Phys. Rev. B 58 213
[45]	Luo L and Tang L H, in preparation

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