General Scaling Relations in Anomalous Diffusion (original) (raw)
Related papers
Analytical results for long-time behavior in anomalous diffusion
2012
We investigate through a Generalized Langevin formalism the phenomenon of anomalous diffusion for asymptotic times, and we generalized the concept of the diffusion exponent. A method is proposed to obtain the diffusion coefficient analytically through the introduction of a time scaling factor λ. We obtain as well an exact expression for λ for all kinds of diffusion. Moreover, we show that λ is a universal parameter determined by the diffusion exponent. The results are then compared with numerical calculations and very good agreement is observed. The method is general and may be applied to many types of stochastic problem.
Journal of Physics A: Mathematical and Theoretical
Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in natural world. In the CTRW framework, we take the waiting time probability density function (PDF) as an infinite series in three parameter Mittag-Leffler functions. According to the models, the mean squared displacement of the process is analytically obtained and numerically verified, in particular, the trend of its transition is shown; furthermore the stochastic representation of the process is presented and the positiveness of the PDF of the position of the particles is strictly proved. Finally, the fractional moments of the model are calculated, and the analytical solutions of the model with external harmonic potential are obtained and some applications are proposed.
Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity
2019
We show analytically that there is anomalous diffusion when the diffusion constant depends on the concentration as a power law with a positive exponent or a negative exponent with absolute value less than one and the initial condition is a delta function in the concentration. On the other hand, when the initial concentration profile is a step, the profile spreads as the square root of time. We verify our results numerically using particles moving stochastically.
Characterization of diffusion processes: Normal and anomalous regimes
We use the evolution of distributions to characterize normal or anomalous diffusion. • We considered space and time dynamics to test different regimes of diffusion. • The width of the distribution is related to the parameters of the random walk. • This analysis is independent of the mechanism responsible for the diffusion. • The technique is specially useful in superdiffusion, where the variance diverges. a b s t r a c t Many man-made and natural processes involve the diffusion of microscopic particles subject to random or chaotic, random-like movements. Besides the normal diffusion characterized by a Gaussian probability density function, whose variance increases linearly in time, so-called anomalous-diffusion regimes can also take place. They are characterized by a variance growing slower (subdiffusive) or faster (superdiffusive) than normal. In fact, many different underlying processes can lead to anomalous diffusion, with qualitative differences between mechanisms producing subdiffusion and mechanisms resulting in superdiffusion. Thus, a general description, encompassing all three regimes and where the specific mechanisms of each system are not explicit, is desirable. Here, our goal is to present a simple method of data analysis that enables one to characterize a model-less diffusion process from data observation, by observing the temporal evolution of the particle spread. To generate diffusive processes in different regimes, we use a Monte-Carlo routine in which both the step-size and the time-delay of the diffusing particles follow Pareto (inverse-power law) distributions, with either finite or diverging statistical momenta. We discuss on the application of this method to real systems.
Physica D: Nonlinear Phenomena, 1999
The superdiffusion behavior, i.e. < x 2 (t) >∼ t 2ν , with ν > 1/2, in general is not completely characherized by a unique exponent. We study some systems exhibiting strong anomalous diffusion, i.e. < |x(t)| q >∼ t qν(q) where ν(2) > 1/2 and qν(q) is not a linear function of q. This feature is different from the weak superdiffusion regime, i.e. ν(q) = const > 1/2, as in random shear flows.
Numerical study of strong anomalous diffusion
Physica a, 2000
The superdi usion behavior, i.e., x 2 (t) ∼ t 2 , with ¿ 1=2, in general is not completely characterized by a unique exponent. We study some systems exhibiting strong anomalous di usion, i.e., |x(t)| q ∼ t q (q) where (2) ¿ 1=2 and q (q) is not a linear function of q. This feature is di erent from the weak superdi usion regime, i.e., (q) = const ¿ 1=2, as in random shear ows. The strong anomalous di usion can be generated by nontrivial chaotic dynamics, e.g. Lagrangian motion in 2d time-dependent incompressible velocity ÿelds, 2d symplectic maps and 1d intermittent maps. Typically the function q (q) is piecewise linear. This corresponds to two mechanisms: a weak anomalous di usion for the typical events and a ballistic transport for the rare excursions. In order to have strong anomalous di usion one needs a violation of the hypothesis of the central limit theorem, this happens only in a very narrow region of the control parameters space. In the presence of strong anomalous di usion one does not have a unique exponent and therefore one has the failure of the usual scaling of the probability distribution, i.e., P(x; t) = t − F(x=t). This implies that the e ective equation at large scale and long time for P(x; t), can obey neither the usual Fick equation nor other linear equations involving temporal and=or spatial fractional derivatives.
Microscopic dynamics underlying anomalous diffusion
Physical Review E, 2000
The time dependent Tsallis statistical distribution describing anomalous diffusion is usually obtained in the literature as the solution of a non-linear Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347 (1995)]. The scope of the present paper is twofold. Firstly we show that this distribution can be obtained also as solution of the non-linear porous media equation. Secondly we prove that the time dependent Tsallis distribution can be obtained also as solution of a linear FP equation [G. Kaniadakis and P. Quarati, Physica A, 237, 229 (1997)] with coefficients depending on the velocity, that describes a generalized Brownian motion. This linear FP equation is shown to arise from a microscopic dynamics governed by a standard Langevin equation in presence of multiplicative noise. PACS number(s): 05.10.Gg , 05.20.-y
Communication: A scaling approach to anomalous diffusion
The Journal of chemical physics, 2014
The paper presents a rigorous derivation of the velocity autocorrelation function for an anomalously diffusing slow solute particle in a bath of fast solvent molecules. The result is obtained within the framework of the generalized Langevin equation and uses only scaling arguments and identities which are based on asymptotic analysis. It agrees with the velocity autocorrelation function of an anomalously diffusing Rayleigh particle whose dynamics is described by a fractional Ornstein-Uhlenbeck process in velocity space. A simple semi-analytical example illustrates under which conditions the latter model is appropriate.
Anomalous diffusion: A dynamic perspective
Physica A: Statistical Mechanics and its Applications, 1990
This paper investigates whether spontaneous, stationary velocity fluctuations can lead to deviations from the regular Fickian diffusion. A kinematic analysis reveals that anomalous diffusion, both fast and slow, arises from long-tailed velocity auto-correlation functions (VACF). This infinite span of interdependence of the random velocity leads to the breakdown of the central limit theorem for particle displacements. A generalized Langevin equation, which features a retarded friction, has been used to describe the particle dynamics in the long-time limit. The analysis reveals that simple power-law decay models for the friction kernel are adequate to yield the pathological VACFs which imply anomalous diffusion. The fluctuation dissipation theorem is invoked to infer that a fractional noise gives rise to anomalous diffusion. Such a Langevin equation represents a mean-field description of disorder effects and the friction kernel then becomes a constitutive property of the medium.