Subdiffusion-limited

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... 

These equations describe subdiffusion-limited chemical reactions. As stated in Chap. 2, such reactions are outside the scope of this monograph. For results on fronts and Turing instabilities in systems with subdiffusion-controlled chemical reactions, see for example [490,240, 147, 148, 235, 318,191]. 

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 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.

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). 

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... 

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. 

For highly concentrated polymer solutions, FCS measurements revealed subdiffusive motion as an additional mode on an intermediate timescale between the fast collective diffusion and the slow self-diffusion [24]. In such slow systems, however, FCS reaches its limits when probe motion becomes so slow that the number of molecules moving into or out of the confocal volume within the measurement time is too small to allow for reliable statistics. Increasing the measurement time is often not straightforward since all fluorescence dyes have only a limited photostability. If a dye bleaches within the confocal volume, it will fake a faster diffusional motion than its real value. Therefore, for the study of such concentrated systems, wide-field fluorescence microscopy and subsequent single molecule tracking is a much better method [120] and has been utilized to study the glass transition [87, 121]. 

Equation 49 is subdiffusive with y = 1/2 in Eq. 1. This is due to the fact that the trajectory of a bead in a chain is spatially confined by its neighbours. In the limit t tr the chain diffuses as one entity and the bead s trajectory follows mainly the motion of the chain s center of mass. The chain s center of mass obeys... 

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