Geodesic
Continuous Time Random Walks with Reactions Forcing and Trapping
C. N. Angstmann1 ; I. C. Donnelly1 ; B. I. Henry1
1 School of Mathematics and Statistics, University of New South Wales, Sydney, Australia
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 2, pp. 17-27 Cet article a éte moissonné depuis la source EDP Sciences

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One of the central results in Einstein’s theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation.
DOI : 10.1051/mmnp/20138202

C. N. Angstmann 1 ; I. C. Donnelly 1 ; B. I. Henry 1

1 School of Mathematics and Statistics, University of New South Wales, Sydney, Australia 
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C. N. Angstmann; I. C. Donnelly; B. I. Henry. Continuous Time Random Walks with Reactions Forcing and Trapping. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 2, pp. 17-27. doi: 10.1051/mmnp/20138202

[1] A. D. Fokker Annalen der Physik 1914 810 820

[2] M. Planck, Sitzber. Preu. Akad. Wiss., (1917), p. 324.

[3] H. Risken. The Fokker-Planck equation: Methods of solution and applications. Second Edition., vol. 18. Springer Verlag, 1996.

[4] E. Barkai, R. Metzler, J. Klafter Phys. Rev. E 2000 132

[5] I. M. Sokolov, J. Klafter Phys. Rev. Lett. 2006 140602

[6] M. Magdziarz, A. Weron, J. Klafter Phys. Rev. Lett. 2008 210601

[7] B. I. Henry, T. A. M. Langlands, P. Straka Phys. Rev. Lett. 2010 170602

[8] A. Weron, M. Magdziarz, K. Weron Phys. Rev. E 2008 036704

[9] M. G. Hahn, K. Kobayashi, S. Umarov. Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion. Proc. Amer. Math. Soc., (2011), pp. 691-705.

[10] V. P. Shkilev J. Exp. Theor. Phys. 2012 830

[11] B. I. Henry, S. L. Wearne Physica A 2000 448 455

[12] B. I. Henry, T. A. M. Langlands, S. L. Wearne Phys. Rev. E 2006 031116

[13] I. M. Sokolov, M. G. W. Schmidt, F. Sagués Phys. Rev. E 2006 031102

[14] T. A. M. Langlands, B. I. Henry, S. L. Wearne Phys. Rev. E 2008 021111

[15] S. Fedotov Phys. Rev. E 2010 011117

[16] E. Abad, S. B. Yuste, K. Lindenberg Phys. Rev. E 2010 031115

[17] G. Bel, E. Barkai Phys. Rev. Lett. 2005 240602

[18] M. Magdziarz, A. Weron, K. Burnecki, J. Klafter Phys. Rev. Lett. 2009 180602

[19] W. Deng, E. Barkai Phys. Rev. E 2009 011112

[20] A. Weigel, B. Simon, M. Tamkun, D. Krapf Proc. Natl. Acad. Sci. 2011 6438 6443

[21] T. A. M. Langlands, B. I. Henry Phys. Rev. E 2010 051102

[22] S. Fedotov Phys. Rev. E 2011 021110

[23] I. Eliazar, J. Klafter Ann. Phys. 2011 2517 2531

[24] K. Ritchie, X. Y. Shan, J. Kondo, K. Iwasawa, T. Fujiwara, A. Kusumi Biophysical journal 2005 2266

[25] F. Santamaria, S. Wils, E. De Schutter, G. Augustine European Journal of Neuroscience 2011 561 568

[26] M. Saxton Biophysical journal 1996 1250 1262

[27] N. Malchus, M. Weiss J. Fluoresc. 2010 19 26

[28] J. H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, R. Metzler Phys. Rev. Lett. 2011 48103

[29] F. Santamaria, S. Wils, E. De Schutter, G. J. Augustine Neuron 2006 635 648

[30] F. Santamaria, S. Wils, E. De Schutter, G. J. Augustine Eur. J. Neurosci. 2011 561 568

[31] B. I. Henry, T. A. M. Langlands, S. L. Wearne Phys. Rev. Lett. 2008 128103

[32] T. A. M. Langlands, B. I. Henry, S. L. Wearne J. Math. Biol., vol. 2009 761 808

[33] T. A. M. Langlands, B. I. Henry, S. L. Wearne SIAM J. Appl. Math. 2011 1168 1203

[34] A. Lubelski, J. Klafter Biophys. J. 2008 4646 4653

[35] A. Kolmogoroff, I. Petrovsky, N. Piscounoff Moscow Univ. Bull. Math 1937 1 25

[36] R. Fisher Ann. Hum. Genet. 1937 355 369

[37] D. Ben-Avraham, S. Havlin. Diffusion and reactions in fractals and disordered systems. Cambridge University Press, 2000.

[38] M. O. Vlad, J. Ross Phys. Rev. E 2002 061908

[39] C. N. Angstmann, I. C. Donnelly, B. I. Henry Phys. Rev. E 2012 032804

[40] S. Fedotov, S. Falconer Phys. Rev. E 2012 031132

[41] H. Scher, M. Lax Phys. Rev. B 1973 4491 4502

[42] A. Yadav, W. Horsthemke Phys. Rev. E 2006 066118

[43] A. V. Chechkin, R. Gorenflo, I. M. Sokolov J. Phys. A 2005 L679

[44] B. Berkowitz, A. Cortis, M. Dentz, H. Scher Rev. Geophys. 2006 RG2003

[45] E. Scalas, R. Gorenflo, F. Mainardi, M. Raberto Fractals 2003 281 289

[46] T. H. Hildebrandt The Amer. Math. Monthly 1938 265

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