Chemistry, Physics and Technology of Surface, 2016, 7 (2), 175-185.

Determination of the parameters of subdiffusion equation based on SPT-experiment data



DOI: https://doi.org/10.15407/hftp07.02.175

V. P. Shkilev, V. V. Lobanov

Abstract


A method is proposed of determining the subdiffuson equation parameters. The method is based on a single particle tracking experiment data analysis. The combination of random barriers model and random traps model is considered as a model of subdiffusion. Subdiffusion equation related to this model contains two memory functions as a parameters. The first function characterizes the distribution of the heights of barriers, the other one characterizes the distribution of the depths of traps. Within the framework of the model under consideration, expressions are derived for the mean square displacement of particles averaged over time and over the ensemble. Expressions are derived for random walks in unlimited space and in a restricted domain of arbitrary shape with the reflectiing boundary. It has been shown that the derived expressions allow us to determine the memory functions appearing in subdiffusion equation. For this purpose, single particle tracking experiment data should be approximated by using these expressions. The method proposed can be used for modeling the diffusion controlled reactions in disordered media.

Keywords


stationary subdiffusion; transient subdiffusion; multiple trapping model; the model of random barriers; mean square displacement

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References


1. Metzler R., Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 2004. 37(31): R161. https://doi.org/10.1088/0305-4470/37/31/R01

2. Klages R., Radons G., Sokolov I.M. Anomalous Transport: Foundations and Applications. (Weinheim: Wiley-VCH, 2007).

3. Mendez V., Fedotov S., Horsthemke W. Reaction-Transport Systems: Mesoscopic foundation, Fronts, and Spatial Instabilities. (Berlin: Springer-Ferlag, 2010). https://doi.org/10.1007/978-3-642-11443-4

4. Meroz Y., Sokolov I.M., Klafter J. Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist. Phys. Rev. E. 2010. 81: R010101. https://doi.org/10.1103/physreve.81.010101

5. Weigel A.V., Simon B., Tamkun M.M., Krapf D. Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking. Proc. Nat. Acad. Sci . 2011. 108: 6438. https://doi.org/10.1073/pnas.1016325108

6. Shkilev V.P. Subdiffusion of mixed origin with chemical reactions. J. Exp. Theor. Phys. 2013. 117(6): 1066. https://doi.org/10.1134/S1063776113140045

7. Schirmacher W. Microscopic theory of dispersive transport in disordered semiconductors. Solid State Commun. 1981. 39: 893. https://doi.org/10.1016/0038-1098(81)90032-6

8. Movaghar B., Grunewald M., Pohlmann B., Wurtz D., Schirmacher W. Theory of hopping and multiple-trapping in disordered systems. J. Stat. Phys. 1983. 30: 315. https://doi.org/10.1007/BF01012306

9. Godzik K., Schirmacher W. Theory of dispersive transport in amorphous semiconductors. J. de Phys. (Paris). 1981. 42: 127. https://doi.org/10.1051/jphyscol:1981424

10. Barkai E., Cheng Y. Aging continuous time random walks. J. Chem. Phys. 2003. 118: 6167. https://doi.org/10.1063/1.1559676

11. Sokolov I.M. Thermodynamics and fractional Fokker-Planck equations. Phys. Rev. E. 2001. 63: R056111.https://doi.org/10.1103/physreve.63.056111

12. Sokolov I.M. Solutions of a class of non-Markovian Fokker-Planck equations. Phys. Rev. E. 2002. 66: R041101. https://doi.org/10.1103/physreve.66.041101

13. Neusius T., Sokolov I.M., Smith J.C. Subdiffusion in time-averaged, confined random walks. Phys. Rev. E. 2009. 80: R011109. https://doi.org/10.1103/physreve.80.011109

14. Miyaguchi T., Akimoto T. Ergodic properties of continuous-time random walks: Finite-size effects and ensemble dependences. Phys. Rev. E. 2013. 87: R032130. https://doi.org/10.1103/physreve.87.032130 

15. Bateman H., Erdelyi A. Tables of Integral Transforms. (NY: Mc Graw-Hill, 1954).Feller W. An

16. Introduction to Probability Theory and its Application. (NY: J. Wiley and Song. Inc., 1957).




DOI: https://doi.org/10.15407/hftp07.02.175

Copyright (©) 2016 V. P. Shkilev, V. V. Lobanov

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