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Continuous Time Random Walks with Reactions Forcing and Trapping

Published online by Cambridge University Press:  24 April 2013

C. N. Angstmann ,
I. C. Donnelly  and
B. I. Henry
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Abstract

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.

Keywords

fractional diffusionreaction-diffusionrandom walkFokker-Planck equationstochastic process
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 17 - 27
Copyright
© EDP Sciences, 2013

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References

Fokker, A. D.. Die mittlere energie rotierender elektrischer dipole im strahlungsfeld. Annalen der Physik, vol. 348, no. 5, (1914), pp. 810-820. CrossRefGoogle Scholar
M. Planck, Sitzber. Preu. Akad. Wiss., (1917), p. 324.
H. Risken. The Fokker-Planck equation: Methods of solution and applications. Second Edition., vol. 18. Springer Verlag, 1996.
Barkai, E., Metzler, R., Klafter, J.. From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E, vol. 61, no. 1, (2000), p. 132. CrossRefGoogle ScholarPubMed
Sokolov, I. M., Klafter, J.. Field-induced dispersion in subdiffusion. Phys. Rev. Lett., vol. 97, no. 14, (2006), p. 140602. CrossRefGoogle Scholar
Magdziarz, M., Weron, A., Klafter, J.. Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: The case of a time-dependent force. Phys. Rev. Lett., vol. 101, no. 21, (2008), p. 210601. CrossRefGoogle ScholarPubMed
Henry, B. I., Langlands, T. A. M., Straka, P.. Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces. Phys. Rev. Lett., vol. 105, no. 17, (2010), p. 170602. CrossRefGoogle ScholarPubMed
Weron, A., Magdziarz, M., Weron, K.. Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation. Phys. Rev. E, vol. 77, no. 3, (2008), p. 036704. CrossRefGoogle Scholar
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.
Shkilev, V. P.. Subdiffusion in a time-dependent force field. J. Exp. Theor. Phys., vol. 114, (2012), p. 830. CrossRefGoogle Scholar
Henry, B. I., Wearne, S. L.. Fractional reaction-diffusion. Physica A, vol. 276, no. 3, (2000), pp. 448-455. CrossRefGoogle Scholar
Henry, B. I., Langlands, T. A. M., Wearne, S. L.. Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations. Phys. Rev. E, vol. 74, no. 3, (2006), p. 031116. CrossRefGoogle ScholarPubMed
Sokolov, I. M., Schmidt, M. G. W., Sagués, F.. Reaction-subdiffusion equations. Phys. Rev. E, vol. 73, no. 3, (2006), p. 031102. CrossRefGoogle ScholarPubMed
Langlands, T. A. M., Henry, B. I., Wearne, S. L.. Anomalous subdiffusion with multi-species linear reaction dynamics. Phys. Rev. E, vol. 77, no. 2, (2008), p. 021111. CrossRefGoogle Scholar
Fedotov, S.. Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts. Phys. Rev. E, vol. 81, no. 1, (2010), p. 011117. CrossRefGoogle ScholarPubMed
Abad, E., Yuste, S. B., Lindenberg, K.. Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous-time random walks. Phys. Rev. E, vol. 81, no. 3, (2010), p. 031115. CrossRefGoogle ScholarPubMed
Bel, G., Barkai, E.. Weak ergodicity breaking in the continuous-time random walk. Phys. Rev. Lett., vol. 94, no. 24, (2005), p. 240602. CrossRefGoogle Scholar
Magdziarz, M., Weron, A., Burnecki, K., Klafter, J.. Fractional Brownian motion versus the continuous-time random walk: A simple test for subdiffusive dynamics. Phys. Rev. Lett., vol. 103, (2009), p. 180602. CrossRefGoogle ScholarPubMed
Deng, W., Barkai, E.. Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E, vol. 79, no. 1, (2009), p. 011112. CrossRefGoogle ScholarPubMed
Weigel, A., Simon, B., Tamkun, M., Krapf, D.. Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking. Proc. Natl. Acad. Sci., vol. 108, no. 16, (2011), pp. 6438-6443. CrossRefGoogle ScholarPubMed
Langlands, T. A. M., Henry, B. I.. Fractional chemotaxis diffusion equations. Phys. Rev. E, vol. 81, no. 5, (2010), p. 051102. CrossRefGoogle ScholarPubMed
Fedotov, S.. Subdiffusion, chemotaxis, and anomalous aggregation. Phys. Rev. E, vol. 83, no. 2, (2011), p. 021110. CrossRefGoogle Scholar
Eliazar, I., Klafter, J.. Anomalous is ubiquitous. Ann. Phys., vol. 326, no. 9, (2011), pp. 2517-2531. CrossRefGoogle Scholar
Ritchie, K., Shan, X. Y., Kondo, J., Iwasawa, K., Fujiwara, T., Kusumi, A.. Detection of non-Brownian diffusion in the cell membrane in single molecule tracking. Biophysical journal, vol. 88, no. 3, (2005), p. 2266. CrossRefGoogle ScholarPubMed
Santamaria, F., Wils, S., De Schutter, E., Augustine, G.. The diffusional properties of dendrites depend on the density of dendritic spines. European Journal of Neuroscience, vol. 34, no. 4, (2011), pp. 561-568. CrossRefGoogle Scholar
Saxton, M.. Anomalous diffusion due to binding: a monte carlo study. Biophysical journal, vol. 70, no. 3, (1996), pp. 1250-1262. CrossRefGoogle Scholar
Malchus, N., Weiss, M.. Elucidating anomalous protein diffusion in living cells with fluorescence correlation spectroscopy-facts and pitfalls. J. Fluoresc., vol. 20, (2010), pp. 19-26. CrossRefGoogle ScholarPubMed
Jeon, J. H., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C., Berg-Sørensen, K., Oddershede, L., Metzler, R., In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett., vol. 106, no. 4, (2011), p. 48103. CrossRefGoogle ScholarPubMed
Santamaria, F., Wils, S., De Schutter, E., Augustine, G. J.. Anomalous diffusion in Purkinje cell dendrites caused by spines. Neuron, vol. 52, no. 4, (2006), pp. 635-648. CrossRefGoogle ScholarPubMed
Santamaria, F., Wils, S., De Schutter, E., Augustine, G. J.. The diffusional properties of dendrites depend on the density of dendritic spines. Eur. J. Neurosci., vol. 34, no. 4, (2011), pp. 561-568. CrossRefGoogle ScholarPubMed
Henry, B. I., Langlands, T. A. M., Wearne, S. L.. Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett., vol. 100, no. 12, (2008), p. 128103. CrossRefGoogle ScholarPubMed
Langlands, T. A. M., Henry, B. I., Wearne, S. L.. Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions J. Math. Biol., vol. 59, no. 6, (2009), pp. 761-808. CrossRefGoogle ScholarPubMed
Langlands, T. A. M., Henry, B. I., Wearne, S. L.. Fractional cable equation models for anomalous electrodiffusion in nerve cells: Finite domain solutions. SIAM J. Appl. Math., vol. 71, no. 4, (2011), pp. 1168-1203. CrossRefGoogle Scholar
Lubelski, A. Klafter, J.. Fluorescence recovery after photobleaching: the case of anomalous diffusion. Biophys. J., vol. 94, no. 12, (2008), pp. 4646-4653. CrossRefGoogle ScholarPubMed
Kolmogoroff, A., Petrovsky, I., Piscounoff, N.. Étude de l’équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique. Moscow Univ. Bull. Math, vol. 1, (1937), pp. 1-25. Google Scholar
Fisher, R.. The wave of advance of advantageous genes. Ann. Hum. Genet., vol. 7, no. 4, (1937), pp. 355-369. Google Scholar
D. Ben-Avraham, S. Havlin. Diffusion and reactions in fractals and disordered systems. Cambridge University Press, 2000.
Vlad, M. O., Ross, J.. Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of neolithic transition. Phys. Rev. E, vol. 66, no. 6, (2002), p. 061908. CrossRefGoogle ScholarPubMed
Angstmann, C. N., Donnelly, I. C., Henry, B. I.. Pattern formation on networks with reactions: A continuous time random walk approach. Phys. Rev. E, vol. 87, no. 3, (2012), p. 032804. CrossRefGoogle Scholar
Fedotov, S., Falconer, S.. Subdiffusive master equation with space-dependent anomalous exponent and structural insta- bility. Phys. Rev. E, vol 85, no. 3, (2012), p. 031132. CrossRefGoogle Scholar
Scher, H., Lax, M.. Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B, vol. 7, (1973), pp. 4491-4502. CrossRefGoogle Scholar
Yadav, A., Horsthemke, W.. Kinetic equations for reaction-subdiffusion systems: Derivation and stability analysis. Phys. Rev. E, vol. 74, no. 6, (2006), p. 066118. CrossRefGoogle ScholarPubMed
Chechkin, A. V., Gorenflo, R., Sokolov, I. M.. Fractional diffusion in inhomogeneous media. J. Phys. A, vol. 38, (2005), p. L679. CrossRefGoogle Scholar
Berkowitz, B., Cortis, A., Dentz, M., Scher, H.. Modelling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys., vol. 44, (2006), p. RG2003. CrossRefGoogle Scholar
Scalas, E., Gorenflo, R., Mainardi, F., Raberto, M.. Revisiting the derivation of the fractional diffusion equation. Fractals, vol. 11, (2003), pp. 281-289. CrossRefGoogle Scholar
Hildebrandt, T. H.. Definitions of Stieltjes integrals of the Riemann type. The Amer. Math. Monthly, vol. 45, (1938), p. 265. CrossRefGoogle Scholar