Subdiffusive transport

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. 

The memory kernel in (2.59), recall that v] represents a nonlocal-in-time integral operator, is a clear indication that subdiffusive transport is non-Markovian. Incorporating kinetic terms into a non-Markovian transport equation requires great care and is best carried out at the mesoscopic level. We show in Sect. 3.4 how to proceed directly at the level of the mesoscopic balance equations for non-Markovian CTRWs. Here we pursue a different approach. As stated above, if all processes are Markovian, then contributions from different processes are indeed separable and simply additive. As is well known, processes often become Markovian if a sufficiently large and appropriate state space is chosen. For the case of reactions and subdiffusion, the goal of a Markovian description can be achieved by taking the age structure of the system explicitly into account as done by Vlad and Ross [460,461]. This approach is equivalent to Model B, see Sect. 3.4. 

Remark 2.4 In the derivation of the generalized reaction-diffusion equation (2.82) we do not explicitly refer to the particular form of the waiting time PDF. Equation (2.82) is valid for arbitrary waiting time PDFs < (t) and has much wider applicability than subdiffusive transport. 

The generalized reaction-diffusion equation (2.82) can be written in a form using fractional derivatives for subdiffusive transport, where the waiting PDF of species i is given in Laplace space by (2.52), (i) 1 —. In that case... 

In this section we show how to obtain subdiffusive transport by using the idea of inverse subordination [278, 371]. Assume that the density 0 obeys the Kolmogorov-Feller equation... 

At 0 < p < 1 it is said about subdiffusive transport processes, at 1 < < 2 - about superdiffusive ones and p = 1 corresponds to classical (Gaussian) diffusion. In its turn, the exponent p is coimected with Hurst ejqtonent H by the equation [42] ... 

The values i, = 0.099-0.988 quoted above suppose that at the curing of system 2DPP+HCE/DDM only slow diffusion (subdiffusive transport) is realised and reaching T = 513 K of the condition d.= d = d means that in this case classical Gaussian... 

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... 

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). 

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. 

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... 

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] ... 

Reactions and Transport Diffusion, Inertia, tind Subdiffusion... 

As shown above, the standard diffusion equation (2.1) has a fractional diffusion equation (2.59) as its analog in the subdiffusive case. As in the case of reaction-transport equation with inertia, see Sect. 2.2, the question arises how to combine reactions and subdiffusion in the activation-controlled regime. (For a discussion of the subdiffusion-limited case, which is outside the scope of this monograph as mentioned on page 34, see for example [491-493, 369, 391, 392, 389, 409, 410, 390, 411, 203, 187].) In some schemes, [188, 189, 186, 187], reactions terms are simply added to the fractional diffusion equation, in a manner similar to the ad hoc HRDEs (2.16), assuming at the outset that the effects of subdiffusion and reactions are separable as in the standard reaction-diffusion (2.11). However, it is easy to... 

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