Simulation and Calibration Between Parameters of Continuous Time Random Walks and Subdifusive Model
DOI:
https://doi.org/10.5540/tema.2017.018.02.0305Palavras-chave:
Anomalous diffusion, fractional diffusion equation, calibrationResumo
We address the problem of subdiusion or normal diusion to perform a calibration between the parameters used in simulation and the parameters of a subdifusive model. The theoretical model is written as a generalized diusion equation with fractional derivatives in time. The data is generated by simulations consisting of continuous-time random walks with controlled mean waiting time and jump length variance to provide a full range of cases between subdiusion and
normal diusion. From the simulations, we compare the accuracy of two methods to obtain the diusion constant, the order of fractional derivatives: the analysis of the dispersion of the variance in time and the optimization tting of theoretical model solutions to histogram of positions. We highlight the connection between the parameters of the simulations the parameters of the theoretical models.
Referências
A. A. M. Arafa and S. Z. Rida. Exact solutions of fractional-order biological. Communications in Theoretical Physics, 6:992996, 2009.
O. P. Argrawal. Fractional variational calculus in terms of riesz fractional
derivatives. J. Phys. A: Math. Theor., 2007:62876303, 2007.
J. G. Berryman. Evolution of a stable prole for a class of nonlinear diffusion equations with xed boundaries. Journal of mathematical physics, 18:2108 2115, 1977.
M. Bologna, C. Tsallis, and P. Grigolini. Anomalous diusion associated
with nonlinear fractional derivative fokker-planck-like equation: exact timedependent solutions. Physical Review E, 62:2213, 2000.
R. H. Byrd, P. Lu, and J. Nocedal. A limited memory algorithm for bound
constrained optimization. SIAM Journal on Scientic Computing, 16:1190
, 1995.
Gaël Combe, Vincent Richefeu, Marta Stasiak, and Allbens P. F. Atman. Experimental validation of a nonextensive scaling law in conned granular media. Phys. Rev. Lett., 115:238301, Dec 2015.
G. L. Eyink and H. Spohn. Negative-temperature states and large-scale, longlived vortices in two-dimensional turbulence. Journal of statistical physics, 70:833886, 1993.
G. A. Garosi, G. Beke, and M. Schulz. Anomalous diusion and resistivity of a turbulence, weakly ionized plasma. Appl. Phys. Lett., 15:334, 1969.
B. V. Gnedenko and A. N. Kolmogorov. Limit distributions for sums of independent random variables. Bull. Amer. Math. Soc, 62:5052, 1956.
H. J. Haubold, A. M. Mathai, and R. K. Saxena. Mittag-lefler functions and their applications. Journal of Applied Mathematics, 2011:51, 2011.
R. Hilfer, editor. Applications of fractional calculus in physics. World Scientic, 2000.
V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger. Generalized master equations for continuous-times random walks. Journal of Statistical Physics, 9:4550, 1973.
E. K. Lenzi, R. S. Mendes, and C. Tsallis. Crossover in diusion equation: Anomalous and normal behaviors. Physical Review E, 67, 2003.
F. Mainardi. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons & Fractals, 7:14611477, 1996.
F. Mainardi. The fundamental solutions for the fractional diffusion-wave equations. Applied Mathematics Letters, 9(6):2328, 1996.
F. Mainardi. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, 2a ed. edition, 2010.
R. Metzler and J. Klafter. The random walk's guide to anomalous diusion: A fractional dynamics approach. Physics Reports, 339:177, 2000.
R. Metzler and T. F. Nonnenmacher. Space - and time-fracitonal diusion and wave equations, fractional fokker-planck equations, and physical motivation. Chemical Physics, 284:6790, 2002.
E. W. Montroll and G. H. Weiss. Random walks on lattices. J. Math. Phys., II:6 167, 1965.
Paolo Paradisi, Rita Cesari, F. Mainardi, and Francesco Tampieri. The fractional Fick's law for non-local transport processes. Physica A: Statistical Mechanics and its Applications, 293:130142, 2001.
I. Podlubny. Fractional Dierential Equations. Academic Press, 1999.
M. F. Shlesinger and J. Klafter. On the relationship among three theories of relaxation in disordered systems. Proceedings of the National Academy of Sciences, 83:848851, 1986.
J. Stephenson. Some non-linear diffusion equations and fractal diffusion. Physica A, 222:234247, 1995.
L. T. Takahashi and et. al. Mathematical models for the aedes aegypti dispersal dynamics: travelling waves by wing and wind. Bulletin of Matematical Biology, 67:509528, 2005.
C. Tsallis. Introduction to Nonextensive statistical mechanics. Springer, 2009.
C. Tsallis and D. J. Bukman. Anomalous diusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis. Phys. Rev. E, 54:R2197, 1996.
S. Umarov, T. Constantino, and S. Steinberg. On a q-central limit theorem consistent with nonextensive statistical mechanics. Milan Journal of Mathematics, 76:307328, 2008.
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