Diffusion, anomalous

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

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],... 

J. P. Bouchaud, A. Ott, D. Langevin, W. Urbach. Anomalous diffusion in elongated micelles and its Levy flight interpretation. J Phys II (Erance) 2 1465-1482, 1991. 

A. Milchev, K. Binder. Anomalous diffusion and relaxation of collapsed polymer chains. Europhys Lett 26 61 -6l6, 1994. 

Criteria 1-3 are the cardinal characteristics of Fickian diffusion and disregard the functional form of D(ci). Violation of any of these is indicative of non-Fickian mechanisms. Criterion 4 can serve as a check if the D(ci) dependence is known. As mentioned, it is crucial that the sorption curve fully adhere to Fickian characteristics for a valid determination of D from the experimental data. At temperatures well above the glass transition temperature, 7 , Fickian behavior is normally observed. However, caution should be exercised when the experimental temperature is either below or slightly above 7 , where anomalous diffusion behavior often occurs. 

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 this paper we give a brief review of the relation between microscopic dynamical properties and the Fourier law of heat conduction as well as the connection between anomalous conduction and anomalous diffusion. We then discuss the possibility to control the heat flow. 

Recently, a simple formula has been found (Li and Wang, 2003) which connects anomalous heat conductivity with anomalous diffusion. More precisely, it has been shown that for a one dimensional system, if the energy diffusion can be described by... 

For systems with total momentum conservation one typically finds anomalous conductivity, namely the thermal conductivity is divergent with the system size. Anomalous conductivity has been connected with anomalous diffusion via the very simple formula (4). 

Chapter 16 - It is shown, that there is principal difference between the description of generally reagents diffusion and the diffusion defining chemical reaction course. The last process is described within the framework of strange (anomalous) diffusion concept and is controled by active (fractal) reaction duration. The exponent a, defining the value of active duration in comparison with real time, is dependent on reagents structure. 

The form of the time-dependence can be understood from the anomalous diffusion of a piece of Rouse chain, which displaces in time such that rather than t. As a result the exp(-k Dt) scattering from Fickian diffusers is replaced by The initial structure factor S(k,0) is... 

This finding implies a sublinear increase of (r (t)) with time. Thus, the incoherent neutron scattering studies qualify the motion of the hydrogens as an anomalous diffusion involving a sublinearly increasing (r (t)). 

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)...

Hornung G, Berkowitz B, Barkai N (2005) Morphogen gradient formation in a complex environment An anomalous diffusion model. Phys Rev E 72,041916, DOl 10.1103/PhysRev E.72.041916... 

Figure 2. The correlation function G(f), which measures the rate at which a step position anomalously diffuses away from a starting position. This diffusion is limited to the equilibrium width G(f —> oo ) = wi. The time, t is scaled by x t which is the equilibration time (Xe, defined in Eq. (17)) for the case = (perfect sticking). The curves are for (from the top) ...

As seen from Fig. 19.3, a sharp increase of optical density of richlocaine coincides with volume transition of NlPA-AA (see also Fig. 19.1). The percentage of released richlocaine was 44.5% and 7.0% at 40°C and 35°C, respectively. The n value equal to 0.52 at 40 °C reflects the Fickian diffusion while the n=0.26 at 35°C is characteristic of anomalous diffusion of richlocaine [7]. The rate of drug release from NIPA-AA was minimal at pH 5. It gradually increased with increasing pH and leveled off at pH 8 (Fig. 19.5). 

The results show that this concept is capable of long-term anomalous diffusion [17]. Further optimization may lead to matrix-type release systems capable of even longer and more constant drug release. 

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