Subdiffusion

The situation becomes quite different in heterogeneous systems, such as a fluid filling a porous medium. Restrictions by pore walls and the pore space microstructure become relevant if the root mean squared displacement approaches the pore dimension. The fact that spatial restrictions affect the echo attenuation curves permits one to derive structural information about the pore space [18]. This was demonstrated in the form of diffraction-like patterns in samples with micrometer pores [19]. Moreover, subdiffusive mean squared displacement laws [20], (r2) oc tY with y < 1, can be expected in random percolation clusters in the so-called scaling window,... 

The connection between anomalous conductivity and anomalous diffusion has been also established(Li and Wang, 2003 Li et al, 2005), which implies in particular that a subdiffusive system is an insulator in the thermodynamic limit and a ballistic system is a perfect thermal conductor, the Fourier law being therefore valid only when phonons undergo a normal diffusive motion. More profoundly, it has been clarified that exponential dynamical instability is a sufRcient(Casati et al, 2005 Alonso et al, 2005) but not a necessary condition for the validity of Fourier law (Li et al, 2005 Alonso et al, 2002 Li et al, 2003 Li et al, 2004). These basic studies not only enrich our knowledge of the fundamental transport laws in statistical mechanics, but also open the way for applications such as designing novel thermal materials and/or... 

Figure 5. Comparison of prediction (4) with numerical data. Normal diffusion ( ). The ballistic motion ( ). Superdiffusion ID Ehrenfest gas channel (Li et al, 2005)(v) the rational triangle channel (Li et al, 2003) (empty box) the polygonal billiard channel with (i = (V > — 1)7t/4), and 2 = 7r/3 (Alonso et al, 2002)(A) the triangle-square channel gas(Li et al, 2005) (<>) / values are obtained from system size L e [192, 384] for all channels except Ehrenfest channel (Li et al, 2005). The FPU lattice model at high temperature regime (Li et al, 2005) ( ), and the single walled nanotubes at room temperature ( ). Subdiffusion model from Ref. (Alonso et al, 2002) (solid left triangle). The solid curve is f3 = 2 — 2/a.

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

Figure 19 Mean square monomer displacements using the CRC model of PB at three temperatures compared with the monomer displacement in an FRC version of the polymer model. Also indicated is the Rouse-like regime with the subdiffusive t0 61 power law entered after the caging regime (CRC at low T) or after the short time dynamics (FRC and CRC at 353 K).

The data of figure 2 demonstrate, that at the present choice (3=0,25 in reesterification reaction course only antipersistent (subdiffusive) transport processes are possible (a=l is achieved for low-molecular substances with Df= 0 only), i.e., active time is always smaller than real time. This indicates on the important role of Levy flights in strange diffusion type definition. 

Turning to the self-motion of PVE protons, Fig. 5.18 shows the NSE data obtained for Sseif(Q,t). If indeed there existed a second Gaussian subdiffusive regime at length scales shorter than that of the Rouse dynamics, then following Eq. 3.26 also for Q>Qr it should be possible to construct a Q-independent (r t)). This is shown in Fig. 5.19. For Q>0.20 A" a nearly perfect collapse of the experimental data into a single curve is obtained. The time dependence of the determined can be well described by a power law with an exponent of... 

The crossover found between the Rouse and the subdiffusive glassy regime becomes most clear in the Q-dependence of the relaxation times. The correlation functions Schain(Q>0 and Sseif(Q,t) are different and therefore their direct comparison is not possible. However, we may easily relate the characteristic times ... 

Figure 2. Probability density function Q(x, t) for absorbing boundaries in x = 1. Top The subdiffusive case, a = 1/2. Bottom The Brownian case, a = 1. The curves are drawn for the times t — 0.005, 0.1, 10 on the top and for t = 0.05, 0.1, 10 on the bottom. Note the distinct cusp-like shape of the subdiffusive solution in comparison to the smooth Brownian counterpart. For the longest time, the Brownian solution has almost completely decayed.

Figure 3. Survival probability for absorbing boundary conditions positioned at x = 1, plotted for the subdiffusive case a = 1/2 and the Brownian case a = 1 (dashed curve). For longer times, the faster (exponential) decay of the Brownian solution, in comparison to the power-law asymptotic of the Mittag-Leffler behavior, is obvious.

This function has the long-time behavior pa(t) Cat a, where Ca is a constant. The survival probability for the subdiffusive case is plotted in Fig. 3 and compared with the Brownian survival. Clearly, for long times, the survival probability in the subdiffusive system decays in a much slower fashion. 

Figure 5. Mean squared displacement for the fractional (a = 1 /2, full line) and normal (dashed) Omstein-Uhlenbeck process. The Brownian process shows the typical proportionality to t for small times it approaches the saturation value much faster than its subdiffusive analogue, which starts off with the r /2 behavior and approaches the thermal equilibrium value by a power law, compare Eq. (61)...

This model for subdiffusion in the external force field F(x) — — ( ) provides a basis for fractional evolution equations, starting from Langevin dynamics that is combined with long-tailed trapping events possessing a... 

In our presentation, we concentrated on the modeling of subdiffusive phenomena—that is, modeling of processes whose mean squared displacement in the force-free limit follows the power-law dependence (x2 )) oc tK for 0 < k < 1. The extension of fractional dynamics to systems where the transport is subballistic but superdiffiisive, 1 < k < 2, is presently under discussion [77, 78], (compare also Ref. 117). 

Fig. 4 Example trajectories of diffusive (left) and subdiffusive (right) gene carriers recorded with a temporal resolution of 33 ms. The subdiffusive trajectory is characterized by confined motion in the MSD plot. However, the hop diffusion pattern of the trajectory can only be detected by its morphological pattern and not by the shape of MSD plot. Adapted with permission from the American Chemical Society and American Institute of Chemical Engineers [41]...

As gene carriers are internalized by endocytosis, the motion characteristics inside the cell resembles the movement of the endosomal compartments within the cell and the formed vesicles are transported along the microtubule network [38]. Suh et al. [41] quantified the transport of individual internalized polyplexes by multiple-particle tracking and showed that the intracellular transport characteristics of polyplexes depend on spatial location and time posttransfection. Within 30 min, polyplexes accumulated around the nucleus. An average of the transport modes over a 22.5 h period after transfection showed that the largest fraction of polyplexes with active transport was found in the peripheral region of the cells whereas polyplexes close to the nucleus were largely diffusive and subdiffusive. Disruption of the microtubule network by nocodazole completely inhibits active transport and also the perinuclear accumulation of polyplexes [37, 40, 47]. 

This is in contrast to viruses, where the virus particles also show active transport when present in the cytosol after fusion with the plasma membrane or endosomal membrane [60-62], This is due to the ability of specific proteins of the virus particle to bind motor proteins. Single-particle tracking reveals that the quantitative intracellular transport properties of internalized non-viral gene vectors (e.g., polyplexes) are similar to that of viral vectors (e.g., adenovirus) [63]. Suk et al. showed that over 80% of polyplexes and adenoviruses in neurons are subdiffusive and 11-13% are actively transported. However, their trafficking pathways are substantially different. Polyplexes colocalized with endosomal compartments whereas adenovirus particles quickly escaped endosomes after endocytosis. Nevertheless, both exploit the intracellular transport machinery to be actively transported. 

In the special case where the site energies are random fluctuations, this is the Anderson model [20,21]. It is well known that Anderson used this model to prove that randomness makes a crystal become an insulating material. Anderson localization is subtly related to subdiffusion, and consequently this important phenomenon can be interpreted as a form of anomalous diffusion, in conflict with the Markov master equation that is frequently adopted as the generator of ordinary diffusion. It is therefore surprising that this is essentially the same Hamiltonian as that adopted by Zwanzig for his celebrated derivation of the van Hove and, hence, of the Pauli master equation. 

Both the case where the Laplace transform of K(t) of Eq. (24) diverge (superdiffusion) or vanish (subdiffusion) must be treated with caution. These conditions will be the main subject under study in this review. The existence of environment fluctuations makes it possible for us to interpret the electron transport as resulting from random jumps, without involving the notion of wave-function collapse, but this is limited to the case of Poisson statistics. Anderson... 

It is evident that the results of Section IV establishes a nice connection between the GME and the condition of subdiffusion explored with success in the last few years [43]. In fact, the more extended the sojourn time of a random walker on a given site, the slower the resulting diffusion process. Thus, if the exponential waiting time distribution of Eq. (69) is replaced by a slower decay, subdiffusion, namely a diffusion slower than ordinary diffusion, can emerge. 

If we adopt the different walking rule of making the walker travel with constant velocity in between two unpredictable non-Poisson time events, subdiffusion is turned into superdiffusion. The CTRW method can be easily adapted to take care of this different walking rule [42,44]. However, in this case, as we shall see, there does not exist yet an exhaustive approach connecting the CTRW prescriptions to the GME structure discussed in Section III. This means that, not even in principle, it is yet known how to derive this kind of superdiffusion from the conventional Liouville prescriptions of nonequilibrium statistical physics. In this section we plan to make a preliminary illustration of this delicate issue. 

Actually, if we focus our attention on Eq. (180) and we consider the case where the correlation function <1> ( ) has the analytical form of Eq. (148), with 2 < p < 3, we reach the conclusion that the conductivity of the system would become infinite in this case, given the fact that the form of Eq. (148) makes non integrable the correlation function . The current /(f) is the time derivative of ( (f)). Thus, time differentiating Eq. (182) and using Eq. (163), we see that j(t) oc f8 1. Hence the generalized Einstein relation of Eq. (182) is not a problem for subdiffusion [76], since the current tends to vanish in the time asymptotic limit. It becomes a problem for superdiffusion, since in this case the current tends to diverge for t > oo. 

Thus, 8 > 1 yields subdiffusion and 8 < 1 superdiffusion. Both cases imply memory. The condition 8 < 1 implies that a strong fluctuation in the positive direction is most probably followed by a fluctuation in the same direction. The condition 8 > 1, on the contrary, implies that after a positive fluctuation, a fluctuation in the opposite direction ensues. Both conditions are a form of memory. In conclusion, the dynamic approach to FBM is a nice way to create memory, with the memory of the variable being the signature of a coordinated and cooperative process, provided that 8 / 1. 

Let us consider now the case p < 2. In this case the third generalized diffusion equation, derived from a CTRW, is well established [43] as a paradigmatic case of subdiffusion. The correlation function of the... 

An acceptable BQD model could be obtained through subordination. This model would be subordinated to the quantum Zeno model of Section II in the same way as the subdiffusion process of Eq. (318) is subordinated to the ordinary random walk model. This would produce non-Poisson distributions of sojourn times in the light-on and light-off states, and these distributions would be of renewal type [69]. 

Let s consider the physical basis of the change of diffusion regime from slow (subdiffusive) up to rapid (superdiffusive). Anomalous (strange) transport processes are described by the general equation [8] ... 

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