Anomalous diffusion functions

Criteria 1-3 are the cardinal characteristics of Fickian diffusion and disregard the functional form of D(ci). Violation of any of these is indicative of non-Fickian mechanisms. Criterion 4 can serve as a check if the D(ci) dependence is known. As mentioned, it is crucial that the sorption curve fully adhere to Fickian characteristics for a valid determination of D from the experimental data. At temperatures well above the glass transition temperature, 7 , Fickian behavior is normally observed. However, caution should be exercised when the experimental temperature is either below or slightly above 7 , where anomalous diffusion behavior often occurs. 

Fig. 4.16 Time evolution of the mean squared displacement (r ) (empty circle) at 363 K and the non-Gaussian parameter 2 obtained from the simulations at 363 K (filled circle) for the main chain protons of PL The solid vertical arrow indicates the position of the maximum of 2> At times r>r(Qinax)> the crossover time, a2 assumes small values, as in the example shown by the dotted arrows. The corresponding functions (r ) and a2 are deduced from the analysis of the experimental data at 320 K in terms of the jump anomalous diffusion model and are displayed as solid lines for (r )and dashed-dotted lines for a2- (Reprinted with permission from [9]. Copyright 2003 The American Physical Society)...

Figure 2. The correlation function G(f), which measures the rate at which a step position anomalously diffuses away from a starting position. This diffusion is limited to the equilibrium width G(f —> oo ) = wi. The time, t is scaled by x t which is the equilibration time (Xe, defined in Eq. (17)) for the case = (perfect sticking). The curves are for (from the top) ...

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

It has been assumed in Equation (6) that the tracer can freely access all void space, be it infra- or interparticle. Note that if a barrier to this exchange exists instead, the possibility of the onset of anomalous diffusion should be considered.42 In this case, the molecular displacement does not increase linearly as a function of the echo time, due to the physical threshold, which translates in an apparent reduction of the diffusion coefficients (till vanishing) for increasing A. Thus, the independence of De on the echo time must be controlled in order not to produce erratic experimental values. 

We will discuss this state in relation to the recent approaches of the anomalous diffusion theory [31]. It is well known [226-230] that by virtue of the divergent form of Poisson brackets (95) the evolution of the distribution function pip,q t) can be regarded as the flow of a fluid in phase space. Thus the Liouville equation (93) is analogous to the continuity equation for a fluid... 

We take for 9iey(o)) the same function we used in our previous study of aging effects in anomalous diffusion—that is, a function behaving like a power-law of exponent 5 — 1 with 0 < 8 < 2 ... 

We would like to attract the attention of the reader to the case when the environment is a source of anomalous diffusion. Paz et al. [116] studied the decoherence process generated by a supra-ohmic bath, but they did not find any problem with the adoption of the decoherence theory. It is convenient to devote some attention to the case when the fluctuation E, is a source of Levy diffusion [59]. If the fluctuation E, is an uncorrelated Levy process, the characteristic function again decays exponentially, and the only significant change is that the... 

Then, as the first step of approaching the study of self-similarity, we investigate two features that must appear if self-similarity exists power-type distribution function of momenta and anomalous diffusion. We are particularly... 

Anomalous diffusion was first investigated in a one-dimensional chaotic map to describe enhanced diffusion in Josephson junctions [21], and it is observed in many systems both numerically [16,18,22-24] and experimentally [25], Anomalous diffusion is also observed in Hamiltonian dynamical systems. It is explained as due to power-type distribution functions [22,26,27] of trapping and untrapping times of the orbit in the self-similar hierarchy of cylindrical cantori [28]. 

One of the few nontrivial systems for which the presence of anomalous diffusion can be proved rigorously is the 2d random shear flow u = (u(y). 0), where u(y) is a random function [6] such that... 

The two linear regions are associated to two different mechanisms in the diffusion process. For small q s—that is, for the core of the probability distribution function P( Ax, t)—only one exponent (vi = v(q) 0.65 for q < 2) fully characterizes the diffusion process. This means that the typical (i.e., nonrare) events obey a (weak) anomalous diffusion process. Roughly speaking, one can say that at scale l the characteristic time r(/) behaves as x ( ) = On the other hand, for q > 2 the behavior q v(q) q const... 

In the science of complexity the system response X(t) is expected to depart from the totally random condition of the simple random walk model, since such fluctuations are expected to have memory and correlation. In the physics literature, anomalous diffusion has been associated with phenomena with longtime memory such that the autocorrelation function is... 

We note that the characteristic times %int and xef do not exist in anomalous diffusion (a / 1). This is obvious from the properties of the Mittag-Leffler function [see Eqs. (62) and (63)]. However, we shall now demonstrate that the above time constants may also be used to characterize the dynamic susceptibility in anomalous relaxation. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

Our objective is to ascertain how anomalous diffusion modihes the dielectric relaxation in a bistable potential with two nonequivalent wells, Eq. (195). The formal step-off transient solution of Eq. (172) for t > 0 is obtained from the Sturm-Liouville representation, Eq. (179), with the initial (equilibrium) distribution function... 

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