×

Lattice Monte Carlo simulation of Galilei variant anomalous diffusion. (English) Zbl 1351.82075

Summary: The observation of an increasing number of anomalous diffusion phenomena motivates the study to reveal the actual reason for such stochastic processes. When it is difficult to get analytical solutions or necessary to track the trajectory of particles, lattice Monte Carlo (LMC) simulation has been shown to be particularly useful. To develop such an LMC simulation algorithm for the Galilei variant anomalous diffusion, we derive explicit solutions for the conditional and unconditional first passage time (FPT) distributions with double absorbing barriers. According to the theory of random walks on lattices and the FPT distributions, we propose an LMC simulation algorithm and prove that such LMC simulation can reproduce both the mean and the mean square displacement exactly in the long-time limit. However, the error introduced in the second moment of the displacement diverges according to a power law as the simulation time progresses. We give an explicit criterion for choosing a small enough lattice step to limit the error within the specified tolerance. We further validate the LMC simulation algorithm and confirm the theoretical error analysis through numerical simulations. The numerical results agree with our theoretical predictions very well.

MSC:

82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

Software:

mlf; MersenneTwister
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gardiner, C. W., Stochastic Methods: A Handbook for the Natural and Social Science (2009), Springer: Springer Berlin
[2] Klafter, J.; Sokolov, I. M., Anomalous diffusion spreads its wings, Phys. World, 29-32 (2005)
[3] Pekalski, A.; Sznajd-Weron, K., Anomalous Diffusion: From Basics to Applications (1999), Springer: Springer Berlin · Zbl 0909.00059
[4] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[5] Metzler, R.; Klafter, J., The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032
[6] Schwarz, K.; Rieger, H., Efficient kinetic Monte Carlo method for reaction-diffusion problems with spatially varying annihilation rates, J. Comput. Phys., 237, 396-410 (2013) · Zbl 1286.65006
[7] Donev, A.; Bulatov, V. V.; Oppelstrup, T.; Gilmer, G. H.; Sadigh, B.; Kalos, M. H., A first-passage kinetic Monte Carlo algorithm for complex diffusion-reaction systems, J. Comput. Phys., 229, 3214-3236 (2010) · Zbl 1186.82046
[8] Gauthier, M. G.; Slater, G. W., Building reliable lattice Monte Carlo models for real drift and diffusion problems, Phys. Rev. E, 70, 015103R, 1-4 (2004)
[9] Chubynsky, M. V.; Slater, G. W., Optimizing the accuracy of lattice Monte Carlo algorithms for simulating diffusion, Phys. Rev. E, 85, 016709, 1-30 (2012)
[10] Saxton, M. J., Anomalous diffusion due to obstacles: a Monte Carlo study, J. Biophys., 66, 394-401 (1994)
[11] Saxton, M. J., Anomalous diffusion due to binding: a Monte Carlo study, J. Biophys., 70, 1250-1262 (1996)
[12] Saxton, M. J., Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study, J. Biophys., 81, 4, 2226-2240 (2001)
[13] Berry, H.; Chaté, H., Anomalous diffusion due to hindering by mobile obstacles undergoing Brownian motion or Orstein-Ulhenbeck processes, Phys. Rev. E, 89, 022708, 1-9 (2014)
[14] Jeschke, M.; Uhrmacher, A. M., Multi-resolution spatial simulation for molecular crowding, (Proceedings of the 2009 Winter Simulation Conference (2008), IEEE), 1384-1392
[15] Bittig, A. T.; Haack, F.; Maus, C.; Uhrmacher, A. M., Adapting rule-based model descriptions for simulating in continuous and hybrid space, (Proceedings of the 9th International Conference on Computational Methods in Systems Biology (2011), ACM), 161-170
[16] Khan, S.; Reynolds, A. M.; Morrison, I. E.; Cherry, R. J., Stochastic modeling of protein motions within cell membranes, Phys. Rev. E, 71, 041915 (2005)
[17] Metzler, R.; Klafter, J., Subdiffusive transport close to thermal equilibrium: from the Langevin equation to fractional diffusion, Phys. Rev., 61, 6308-6311 (2000)
[18] Saxton, M. J., A biological interpretation of transient anomalous subdiffusion. I. Qualitative model, J. Biophys., 92, 1178-1191 (2007)
[19] Feder, T. J.; Brust-Mascher, I.; Slattery, J. P.; Baird, B.; Webb, W. W., Constrained diffusion or immobile fraction on cell surfaces: a new interpretation, J. Biophys., 70, 2767-2773 (1996)
[20] Nicolau, D. V.; Hancock, J. F.; Burrage, K., Sources of anomalous diffusion on cell membranes: a Monte Carlo study, J. Biophys., 92, 6, 1975-1987 (2007)
[21] Dieterich, P.; Klages, R.; Preuss, R.; Schwab, A., Anomalous dynamics of cell migration, Proc. Natl. Acad. Sci. USA, 105, 2, 459-463 (2008)
[22] Mierke, C. T.; Frey, B.; Fellner, M.; Herrmann, M.; Fabry, B., Integrin \(\alpha 5 \beta 1\) facilitates cancer cell invasion through enhanced contractile forces, J. Cell Sci., 124, 369-383 (2011)
[23] Fedotov, S.; Iomin, A.; Ryashko, L., Non-Markovian models for migration-proliferation dichotomy of cancer cells: anomalous switching and spreading rate, Phys. Rev. E, 84, 061131, 1-8 (2011)
[24] Keren, K.; Yam, P. T.; Kinkhabwala, A.; Mogilner, A.; Theriot, J. A., Intracellular fluid flow in rapidly moving cells, Nat. Cell Biol., 11, 10, 1219-1224 (2009)
[25] Hughes, B. D., Random Walks and Random Environments: Vol. 1. Random Walks (1995), Oxford University Press: Oxford University Press New York · Zbl 0820.60053
[26] Montroll, E. W.; Scher, H., Random walks on lattices: IV. Continuous-time walks and influence of absorbing boundaries, J. Stat. Phys., 9, 2, 101-139 (1973)
[27] Redner, S., A Guide to First-Passage Processes (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0980.60006
[28] Vigelius, M.; Meyer, B., Multi-dimensional, mesoscopic Monte Carlo simulations of inhomogeneous reaction-drift-diffusion systems on Graphics-Processing Units, PLoS ONE, 7, 4, 1-13 (2012)
[29] Shlesinger, M. F.; Klafter, J.; Wong, Y. M., Random walks with infinite temporal and spatial moments, J. Stat. Phys., 27, 499-512 (1982) · Zbl 0521.60080
[30] Klafter, J.; Blumen, A.; Shlesinger, M. F., Stochastic pathway to anomalous diffusion, Phys. Rev. A, 35, 7, 3081-3086 (1987)
[31] Bezzola, A.; Bales, B. B.; Alkire, R. C.; Petzold, L. R., An exact and efficient First Passage Time algorithm for Reaction-Diffusion Processes on a 2D-Lattice, J. Comput. Phys., 256, 183-197 (2014) · Zbl 1349.65006
[32] Oppelstrup, T.; Bulatov, V. V.; Donev, A.; Kalos, M. H.; Gilmer, G. H.; Sadigh, B., First-passage kinetic Monte Carlo method, Phys. Rev. E, 80, 066701, 1-35 (2009)
[33] Oppelstrup, T.; Bulatov, V. V.; Gilmer, G. H.; Kalos, M. H.; Sadigh, B., First-passage Monte Carlo algorithm: diffusion without all the hops, Phys. Rev. Lett., 97, 230602, 1-4 (2006)
[34] de Haan, H. W.; Gauthier, M. G.; Chubynsky, M. V.; Slater, G. W., The importance of introducing a waiting time for Lattice Monte Carlo simulations of a polymer translocation process, Comput. Phys. Commun., 182, 29-32 (2011) · Zbl 1219.82036
[35] Rangarajan, G.; Ding, M., First passage time problem for biased continuous-time random walks, Fractals, 8, 2, 139-145 (2000) · Zbl 1046.82013
[36] Fa, K. S.; Lenzi, E. K., Anomalous diffusion, solutions, and first passage time: influence of diffusion coefficient, Phys. Rev. E, 71, 012101, 1-4 (2005)
[37] Rangarajan, G.; Ding, M., Anomalous diffusion and the first passage time problem, Phys. Rev. E, 62, 1, 120-133 (2000) · Zbl 1050.82545
[38] Morse, P. M.; Feshbach, H., Methods of Theoretical Physics (1953), McGraw-Hill: McGraw-Hill New York, USA · Zbl 0051.40603
[39] Mathai, A. M.; Saxena, R. K.; Haubold, H. J., The H-Function: Theory and Applications (2010), Springer: Springer New York, USA · Zbl 1223.85008
[40] Gitterman, M., Mean first passage time for anomalous diffusion, Phys. Rev. E, 62, 5, 6065-6070 (2000)
[41] Koren, T.; Klafter, J.; Magdziarz, M., First passage times of Lévy flights coexisting with subdiffusion, Phys. Rev. E, 76, 031129, 1-5 (2007)
[42] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series and Products (2007), Academic Press: Academic Press New York · Zbl 1208.65001
[43] Feller, W., An Introduction to Probability Theory and Its Applications, vol. 2 (1971), Wiley · Zbl 0219.60003
[44] Slater, G. W., Theory of band broadening for DNA gel electrophoresis and sequencing, Electrophoresis, 14, 1-7 (1993)
[45] Takahashi, K.; Arjunan, S. N.V.; Tomita, M., Space in systems biology of signaling pathways - towards intracellular molecular crowding in silico, FEBS Lett., 579, 1783-1788 (2005)
[46] Podlubny, I.; Kacenak, M., The Matlab mlf code, MATLAB Central File Exchange, 2012
[47] Rodríguez, J. V.; Kanndorp, J. A.; Dorbrzynski, M.; Blom, J. G., Spatial stochastic modelling of the phosphoenolpyruvate dependent phosphotransferase (PTS) pathway in Escherichia coli, Bioinformatics, 22, 15, 1895-1901 (2006)
[48] Stundzia, A. B.; Lumsden, C. J., Stochastic simulation of coupled reaction-diffusion processes, J. Comput. Phys., 127, 196-207 (1996) · Zbl 0860.65122
[49] Shanno, D. F.; Franta, W. R.; Maly, K., An efficient data structure for the simulation event set, Commun. ACM, 20, 8, 596-602 (1977) · Zbl 0356.68052
[50] McCormack, W. M.; Sargent, R. G., Analysis of future event set algorithms for discrete event simulation, Commun. ACM, 24, 12, 801-812 (1981)
[51] Gorenflo, R.; Loutchko, J.; Luchko, Y., Computation of the Mittag-Leffler function \(E_{\alpha, \beta}\)(z) and its derivative, Fract. Calc. Appl. Anal., 5, 4, 491-518 (2002) · Zbl 1027.33016
[52] Matsumoto, M.; Nishimura, T., Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Trans. Model. Comput. Simul., 8, 1, 3-30 (1998) · Zbl 0917.65005
[53] Jeschke, M.; Ewald, R.; Uhmacher, A. M., Exploring the performance of spatial stochastic simulation algorithms, J. Comput. Phys., 7, 2562-2574 (2011) · Zbl 1316.65012
[54] Nobile, M. S.; Cazzaniga, P.; Besozzi, D.; Pescini, D.; Mauri, G., cuTauLeaping: a GPU-powered tau-leaping stochastic simulator for massive parallel analyses of biological systems, PLoS ONE, 9, 3, 1-20 (2014)
[55] Navarro, C. A.; Kahler, N. H.; Mateu, L., A survey on parallel computing and its applications in data-parallel problems using GPU architectures, Commun. Comput. Phys., 15, 2, 285-329 (2014) · Zbl 1388.65212
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.