Anomalous diffusion mean-square displacement

FIG. 2 Mean-square displacement (MSD) of helium atoms dissolved in polyisobutylene. There is a regime of anomalous diffusion (MSD a followed by a crossover at 100 ps to normal (Einstein) diffusion (MSD a r) [24],... 

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)...

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)...

In the previous section we analyzed the random walk of molecules in Euclidean space and found that their mean square displacement is proportional to time, (2.5). Interestingly, this important finding is not true when diffusion is studied in fractals and disordered media. The difference arises from the fact that the nearest-neighbor sites visited by the walker are equivalent in spaces with integer dimensions but are not equivalent in fractals and disordered media. In these media the mean correlations between different steps (UjUk) are not equal to zero, in contrast to what happens in Euclidean space cf. derivation of (2.6). In reality, the anisotropic structure of fractals and disordered media makes the value of each of the correlations u-jui structurally and temporally dependent. In other words, the value of each pair u-ju-i-- depends on where the walker is at the successive times j and k, and the Brownian path on a fractal may be a fractal of a fractal [9]. Since the correlations u.juk do not average out, the final important result is (UjUk) / 0, which is the underlying cause of anomalous diffusion. In reality, the mean square displacement does not increase linearly with time in anomalous diffusion and (2.5) is no longer exact. 

The type of the random walk (recurrent or nonrecurrent) determines the minimum value of the two terms in the brackets of the previous equation. If the walker does not visit the same sites (nonrecurrent) then dw = 2df/ds. If the walk is of recurrent type then the walker visits the same sites again and again and therefore the walker covers the available space (space-filling walk). Consequently, the meaning of dw coincides with df (dw = df). The mean square displacement in anomalous diffusion follows the pattern... 

Before closing this chapter we would like to mention briefly a novel consideration of diffusion based on the recently developed concepts of fractional kinetics [29]. From our previous discussion it is apparent that if ds < 2, diffusion is recurrent. This means that diffusion follows an anomalous pattern described by (2.10) the mean squared displacement grows as (z2 (t)) oc t1 with the exponent 7 1. To deal with this, a consistent generalization of the diffusion equation (2.18) could have a fractional-order temporal derivative such as... 

In the fractal porous medium, the diffusion is anomalous because the molecules are considerably hindered in their movements, cf. e.g., Andrade et al., 1997. For example, Knudsen diffusion depends on the size of the molecule and on the adsorption fractal dimension of the catalyst surface. One way to study the anomalous diffusion is the random walk approach (Coppens and Malek, 2003). The mean square displacement of the random walker (R2) is not proportional to the diffusion time t, but rather scales in an anomalous way ... 

Experimentally, the effective temperature of a colloidal glass can be determined by studying the anomalous drift and diffusion properties of an immersed probe particle. More precisely, one measures, at the same age of the medium, on the one hand, the particle mean-square displacement as a function of time, and, on the other hand, its frequency-dependent mobility. This program has recently been achieved for a micrometric bead immersed in a glassy colloidal suspension of Laponite. As a result, both Ax2(t) and p(co) are found to display power-law behaviors in the experimental range of measurements [12]. 

Latora et al. [18] discussed a relation between the process of relaxation to equilibrium and anomalous diffusion in the HMF model by comparing the time series of the temperature and of the mean-squared displacement of the phases of the rotators. They showed that anomalous diffusion changes to a normal diffusion after a crossover time, and they also showed that the crossover time coincides with the time when the canonical temperature is reached. They also claim that anomalous diffusion occurs in the quasi-stationary states. 

Figure 33.3 Different results of motion picture with different video frame speeds for the identical trajectory of random walk. In contrast to the case of normal diffusion, observed mean square displacements (MSD) significantly depend on the frame speed in the case of anomalous diffusion.

Anomalous diffusion is also possible in microporous solids. For instance, it is possible for molecules to be confined in a channel system in which they cannot pass each other, and this will obviously affect molecular displacement in a time interval. This case is termed single-file diffusion , and the mean square displacement in a time t is then given... 

For a large variety of applications, simple Brownian motion or Fickian diffusion is not a satisfactory model for spatial dispersal of particles or individuals. Physical, chemical, biological, and ecological systems often display anomalous diffusion, where the mean square displacement (MSD) of a particle does not grow linearly with time ... 

As far as transport properties of a fractal structure are concerned, the mean square displacement (MSD) of a particle follows a power law, (r ) where r is the distance from the origin of the random walk and is known as the random walk dimension. In other words, diffusion on fractals is anomalous, see Sect. 2.3. Recall that for normal diffusion in three-dimensional space the MSD is given by (r ) = 6Dt. For fractals, dy, > 2, and the exponent of t in the MSD is smaller than 1. We introduce a dimensionless distance by dividing r by the typical diffusive... 

Santamaria et al. [375] found that the transport of biologically inert particles, fluorescein dextran, in spiny dendrites is very slow compared with standard diffusion. The mean-square displacement is x t)) with y < 1 [298, 379]. The anomalous diffusion appears to be caused by the dendritic spines acting as the traps for the particles. We present here a mesoscopic model for the transport and biochemical reactions inside a population of spines and dendrites [122]. The morphology of spiny dendrites is very complex the distances between the spines and their sizes and shapes are randomly distributed [179, 362]. The model allows us to deal with the morphological diversity of dendritic spines via the transparent formalism of waiting time distributions. 

Anomalous diffusion of hydrogen molecules is analyzed in these stmctures, which is determined by the geometry features of the water framework. Diffusion of hydrogen molecules in the new Cq and sT hydrogen clathrate stmctures is also analyzed. Mean square displacement analysis shows that diffusion is anisotropic and anomalous at nanosecond timescale. 

Another way of observing the anomalous nature of diffusion on a fractal is to find out what becomes of the diffusion coefficient D t), defined from the classic Einstein relation between mean squared displacement and time ... 

In a more recent study, Muller-Plathe, Rogers, and van Gunsteren have pointed out a case of anomalous diffusion in polyisobutylene near room temperature [48], in harmony with the findings by TSA for gas motion in dense polymers [56]. For He in PIB, anomalous behavior could be clearly shown, and the transition to normal diffusion at around 0.1 ns could be captured. The log-log plot that shows this crossover, is reproduced in Fig. 6. For the much slower diffusing O2, the mean-square displacement data were not accurate enough to determine unambiguously if the curve represented diffusive behavior... 

Regular diffusion, better known as Brownian motion, is characterized (in the absence of directed, external fields) by a linear increase of the mean-square displacement with time, see Eq. 60 for the motion of the center of mass. For anomalous diffusion this simple relation does not hold anymore. Then the temporal evolution of the mean-square displacement is non-linear, and at long times often obeys... 

Its value is crucial, because the magnitude and characteristics of mean-squared displacement (normal vs. anomalous diffusion) are sensitive functions of (A ). Regrettably, several practitioners of this method omit (A ) values from their publications. Arizzi [93] noted that an analytic form is available from the X-ray scattering literature [148],... 

For conducting a kinetic Monte Carlo simulation, the most straightforward choice for a network of sorption states and rate constants is that of the original molecular structure. Its key advantage is its one-to-one correspondence with the detailed polymer configuration. However, the small size of a typical simulation box is a disadvantage. For example, in [97] it was observed that anomalous diffusion continued until the root-mean-squared displacement equaled the box size. From this match in length scales, it is not... 

FIG. 6 Mean-squared displacement for methane in glassy atactic polypropylene, from calculations in [91]. The dot-dash line indicates the box size squared. Dotted lines indicate diffusion exponents of n = 1 /2 and n = 1. A turnover from anomalous to Fickian diffusion is predicted over times of 1-5 (is. 

Concerning PCS, the most interesting observables from the simulations are the trajectories of single diffusion molecules or particles, respectively. From these trajectories, the mean square displacements and the autocorrelation functions can be calculated. This way, it can be analyzed how heterogeneity expresses itself in the PCS results, i.e., how anomalous diffusion is averaged over the relevant PCS time and length scales. Also, the question how interactions between dye and polymer chains influence PCS results has been recently addressed using a combination of PCS experiments with simulations [46] (Fig. 14). 

De Gennes [121] considered the problem of anomalous diffusion on fractal networks in an attempt to understand the conductivity threshold of a percolation cluster. In normal diffusion the mean-square displacement, d, is related to the diffusion coefficient, D, according to ... 

Diffusion in polymers can manifest itself in normal or anomalous form depending on the time or displacement length scale probed in the experiment. The criterion for this distinction is the time dependence of the mean squared displacement... 

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