Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-19T06:56:03.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Elucidating the Role of Subdiffusion and Evanescence in the Target Problem: Some Recent Results

Published online by Cambridge University Press:  24 April 2013

E. Abad ,
S. B. Yuste  and
K. Lindenberg
E. Abad*
Affiliation:
Departamento de Física Aplicada, Centro Universitario de Mérida, Universidad de Extremadura, E-06800 Mérida, Spain
S. B. Yuste
Affiliation:
Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
K. Lindenberg
Affiliation:
Department of Chemistry and Biochemistry, and BioCircuits Institute, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0340, USA
*
⋆ Corresponding author. E-mail: eabad@unex.es
Get access

Abstract

We present an overview of recent results for the classic problem of the survival probability of an immobile target in the presence of a single mobile trap or of a collection of uncorrelated mobile traps. The diffusion exponent of the traps is taken to be either γ = 1, associated with normal diffusive motion, or 0 < γ < 1, corresponding to subdiffusive motion. We consider traps that can only die upon interaction with the target and, alternatively, traps that may die due to an additional evanescence process even before hitting the target. The evanescence reaction is found to completely modify the survival probability of the target. Such evanescence processes are important in systems where the addition of scavenger molecules may result in the removal of the majority species, or ones where the mobile traps have a finite intrinsic lifetime.

Keywords

diffusion limited reactionsanomalous diffusionfractional equations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 100 - 113
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abad, E., Yuste, S. B., Lindenberg, K.. Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous-time random walks. Phys. Rev. E 81 (2010), 031115. CrossRefGoogle ScholarPubMed
Abad, E., Yuste, S.B., Lindenberg, K.. Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: A fractional equation approach.. Phys. Rev. E 86 (2012), 061120. CrossRefGoogle ScholarPubMed
M. Abramowitz, I. A. Stegun. Handbook of Mathematical Functions. Dover, New York, 1965.
Barzykin, A. V., Tachiya, M.. Diffusion-influenced reaction kinetics on fractal structures. J. Chem. Phys. 99 (1993), 9591–9597. CrossRefGoogle Scholar
Bénichou, O., Moreau, M., Oshanin, G.. Kinetics of stochastically gated diffusion-limited reactions and geometry of random walk trajectories. Phys. Rev. E 61 (2000), 3388–3406. CrossRefGoogle ScholarPubMed
Berezhkovskii, A. M., Yang, D.-Y., Lin, S. H., Makhnovskii, Yu. A., Sheu, S.-Y.. Smoluchowski-type theory of stochastically gated diffusion-influenced reactions. J. Chem. Phys. 106 (1997), 6985–6998. CrossRefGoogle Scholar
Borrego, R., Abad, E., Yuste, S.B.. Survival probability of a subdiffusive particle in a d-dimensional sea of mobile traps. Phys. Rev. E 80 (2009), 061121. CrossRefGoogle Scholar
Bramson, M., Lebowitz, J. L.. Asymptotic Behavior of Densities in Diffusion-Dominated Annihilation Reactions. Phys. Rev. Lett. 61 (1988), 2397–2400.
Bray, A. J., Blythe, R. A.. Exact Asymptotics for One-Dimensional Diffusion with Mobile Traps. Phys. Rev. Lett. 89 (2002), 150601. CrossRefGoogle ScholarPubMed
Bray, A.J., Majumdar, S. N., Blythe, R. A.. Formal solution of a class of reaction-diffusion models: Reduction to a single-particle problem. Phys. Rev. E 67 (2003), 060102(R). CrossRefGoogle ScholarPubMed
Burlatsky, S. F., Oshanin, G., Ovchinnikov, A. A.. Fluctuation induced kinetics of incoherent excitations quenching. Phys. Lett. A 139 (1989), 245–248. CrossRefGoogle Scholar
L-C. Chen, R. Sun, A Monotonicity Result for the Range of a Perturbed Random Walk. arXiv:1203.1389v2 [math.PR]
Collins, F. C., Kimball, G. E.. Diffusion-controlled reaction rates. J. Colloid. Sci. 4 (1949), 425–437. CrossRefGoogle Scholar
Donsker, D. V., Varadhan, S. R. S.. Asymptotics for the Wiener sausage. Commun. Pure Appl. Math. 32 (1975), 525–565;
Eaves, J. D., Reichman, D. R.. The subdiffusive targeting problem. J. Phys. Chem. B 112 (2008), 4283-4289. CrossRefGoogle ScholarPubMed
J. Franke, S. Majumdar. Survival probability of an immobile target surrounded by mobile traps. J. Stat. Mech. (2012) P05024.
Glarum, S. H.. Dielectric Relaxation of Isoamyl Bromide. J. Chem. Phys. 33 (1960), 639–643. CrossRefGoogle Scholar
Grebenkov, D. S.. Searching for partially reactive sites: Analytical results for spherical targets. J. Chem. Phys. 132 (2010), 034104. CrossRefGoogle ScholarPubMed
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 74 (2006), 031116. CrossRefGoogle ScholarPubMed
den Hollander, F., Shuler, K.E.. Random walks in a random field of decaying traps. J. Stat. Phys. 67 (1992), 13–31. CrossRefGoogle Scholar
B. H. Hughes. Random Walks and Random Environments. Volume 1: Random Walks. Clarendon Press, Oxford, 1995.
Kim, J., Jung, Y., Jeon, J., Lee, S.. Diffusion-influenced radical recombination in the presence of a scavenger. J. Chem. Phys. 104 (1996), 5784–5797. CrossRefGoogle Scholar
Lomholt, M. A., Zaid, I. M., Metzler, R.. Subdiffusion and Weak Ergodicity Breaking in the Presence of a Reactive Boundary. Phys. Rev. Lett. 98 (2007), 200603. CrossRefGoogle ScholarPubMed
A. M. Mathai, R. K. Saxena. The H-function with Applications in Statistics and Other Disciplines. Wiley, New York, 1978.
See ch. 3 in V. Méndez, S. Fedotov, W. Horsthemke. Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities. Springer, Berlin, 2010.
Metzler, R., Klafter, J.. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77. CrossRefGoogle Scholar
Moreau, M., Oshanin, G., Bénichou, O., Coppey, M.. Pascal principle for diffusion-controlled trapping reactions. Phys. Rev. E 67 (2003), 045104(R). CrossRefGoogle ScholarPubMed
Oshanin, G., Bénichou, O., Coppey, M., Moreau, M.. Trapping reactions with randomly moving traps: Exact asymptotic results for compact exploration. Phys. Rev. E 66 (2002), 060101(R). CrossRefGoogle Scholar
Oshanin, G., Vasilyev, O., Krapivsky, P. L., Klafter, J.. Survival of an evasive prey. PNAS 106 (2009), 13696–13701. CrossRefGoogle ScholarPubMed
I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999.
S. A. Rice. Diffusion-limited Reactions. Elsevier. Amsterdam, 1985.
Ruiz-Lorenzo, J. J., Yuste, S. B., Lindenberg, K.. Simulations for trapping reactions with subdiffusive traps and subdiffusive particles. J. Phys.: Condens. Matter. 19 (2007), 065120. Google Scholar
Sano, H., Tachiya, M.. Partially diffusion controlled recombination. J. Chem. Phys. 71 (1979), 1276–1282. CrossRefGoogle Scholar
Seki, K., Shushin, A. I., Wojcik, M., Tachiya, M.. Specific features of the kinetics of fractional-diffusion assisted geminate reactions. J. Phys.: Condens. Matter 19 (2007), 065117. Google Scholar
Seki, K., Wojcik, M., Tachiya, M.. Fractional reaction-diffusion equation. J. Chem. Phys. 119 (2003), 2165–2170. CrossRefGoogle Scholar
Shkilev, V. P.. Subdiffusion with the Disappearance of Particles at the Time of a Jump. J. Exp. Theor. Phys. (Springer) 112 (2011), 1071–1076. CrossRefGoogle Scholar
Shlesinger, M. F., Montroll, E. W.. On the Williams-Watts function of dielectric relaxation. PNAS 81 (1984), 1280–1283. CrossRefGoogle ScholarPubMed
Smoluchowski, M.. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. 92 (1917), 129–168. Google Scholar
Sokolov, I. M., Schmidt, M. G. W., Sagués, F.. Reaction-subdiffusion equations. Phys. Rev. E 73 (2006), 031102;
S. B. Yuste, E. Abad, K. Lindenberg. Reactions in Subdiffusive Media and Associated Fractional Equations in Fractional Dynamics. Recent Advances. J. Klafter, S. C. Lim, and R. Metzler (Eds.). World Scientific, Singapore, 2011.
Yuste, S. B., Lindenberg, K.. Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents. Phys. Rev. E 72 (2005), 061103. CrossRefGoogle ScholarPubMed
Yuste, S. B., Lindenberg, K.. Subdiffusive target problem: Survival probability. Phys. Rev. E 76 (2007), 051114. CrossRefGoogle ScholarPubMed
Yuste, S. B., Oshanin, G., Lindenberg, K., Bénichou, O., Klafter, J.. Survival probability of a particle in a sea of mobile traps: A tale of tails. Phys. Rev. E 78 (2008), 021105. CrossRefGoogle Scholar
Yuste, S. B., Ruiz-Lorenzo, J. J., Lindenberg, K.. Target problem with evanescent subdiffusive traps. Phys. Rev. E 74 (2006), 046119. CrossRefGoogle ScholarPubMed