Subdiffusion diffusion

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.

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. 

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]. 

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. 

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. 

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... 

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] ... 

Figure 10. Plot of log ((R2(t)) - R2(0))) versus log (t) for a cluster of 735 water molecules (circles), cytochrome c (triangles), and cytochrome c hydrated by 400 water molecules (squares). Plotted fit to water data from 0.1 ps to 0.9 ps has a slope of 0.95 0.05, indicating normal diffusion. The slope of the plotted fit to both cytochrome c results from 0.1 ps to 3.0ps is 0.52 0.01, indicating anomalous subdiffusion with exponent v = 0.26.

Here V(<(), t) = pF(t) cos 4> is the potential arising from an external applied electric field F(f). Here, just as with the translational diffusion equation treated in Ref. 7, we consider subdiffusion, 0 < ct < 1 phenomena only. Here, the internal field effects are ignored, which means that the effects of long-range torques due to the interaction between the average moments and the Maxwell fields are not taken into account. Such effects may be discounted for dilute systems in first approximation. Thus, the results obtained here are relevant to situations where dipole-dipole interactions have been eliminated by extrapolation of data to infinite dilution. 

Anomalous rotational diffusion in a potential may be treated by using the fractional equivalent of the diffusion equation in a potential [7], This diffusion equation allows one to include explicitly in Frohlich s model as generalized to fractional dynamics (i) the influence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the influence of the fast (high-frequency) intrawell relaxation modes on the relaxation process. The fractional translational diffusion in a potential is discussed in detail in Refs. 7 and 31. Here, just as the fractional translational diffusion treated in Refs. 7 and 31, we consider fractional rotational subdiffusion (0rotation about fixed axis in a potential Vo(< >)- We suppose that a uniform field Fi (having been applied to the assembly of dipoles at a time t = oo so that equilibrium conditions prevail by the time t = 0) is switched off at t = 0. In addition, we suppose that the field is weak (i.e., pFj linear response condition). 

At longer times, monomers participate in collective motion of larger sections with smaller effective diffusion coefficient D(t). Therefore the mean-square displacement of monomers is not a linear function of time, but Instead subdiffusive ---------------------------------------------... 

Consistent with the fact that the longest relaxation time of the Zimm model is shorter than the Rouse model, the subdiffusive monomer motion of the Zimm model [(Eq. (8.70)] is always faster than in the Rouse model [Eq. (8.58)] with the same monomer relaxation time tq. This is demonstrated in Fig. 8.8, where the mean-square monomer displacements predicted by the Rouse and Zimm models are compared. Each model exhibits subdiffusive motion on length scales smaller than the size of the chain, but motion becomes diffusive on larger scales, corresponding to times longer than the longest relaxation time. ... 

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