Reactions and Anomalous Diffusion

For n-variable systems, the evolution equations for persistent random walks with reaction read, i = 1, 

For a large variety of applications, simple Brownian motion or Fickian diffusion is not a satisfactory model for spatial dispersal of particles or individuals. Physical, chemical, biological, and ecological systems often display anomalous diffusion, where the mean square displacement (MSD) of a particle does not grow linearly with time  

If 0 y 1, the process is subdiffiisive if y 1, it is superdiffusive. Superdiffusion is encountered, for example, in turbulent fluids [407], in chaotic systems [51], in rotating flows [418, 472], in oceanic gyres [44], for nanorods at viscous interfaces [93], and for surfactant diffusion in living polymers [14]. Subdiffusion is observed in disordered ionic chains [45], in porous systems [100], in amorphous semiconductors [383, 174], in disordered materials [307], in subsurface hydrology [43, 38,23,42,382,91], and for proteins and lipids in plasma membranes of various cells [380, 477, 387], for mRNA molecules in Escherichia coli cells [162], and for proteins in the nucleus [463]. 

Motor proteins can lead to superdiffusive transport of engulfed microbeads in living eukaryotic cells with y = 1.47 0.07 for short times, up to the order of 

Anomalous diffusion is often caused by memory effects and Levy-type statistics [185, 53], Specifically, superdiffusion is observed for random walks with heavytailed jump length distributions and subdiffusion for heavy-tailed waiting time distributions, see Sect. 3.4. The latter type of distribution can be caused by traps that have an infinite mean waiting time [185]. For reviews of anomalous diffusion see, e.g., [298,299, 229]. 

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The results of the study in HA solution suggest a simple but secure strategy to control the reaction by anomalous diffusion occurring in biological systems. The simple polymer solution used here is not only a model for ECM but also one to be extended to other biological space such as cytoplasm and membranes. 

Standard reaction-diffusion equation and introduce two deviations from normal diffusion, namely transport with inertia and anomalous diffusion. We present a phenomenological approach of standard diffusion, transport with inertia, and anomalous diffusion. This chapter also contains a first mesoscopic description of the transport in terms of random walk models. We strongly recommend such a mesoscopic approach to ensure that the reaction-transport equations studied are physically and mathematically sound. We present a comprehensive review of the mesoscopic foundations of reaction-transport equations in Chap. 3, which is at the heart of Part I. 

Part III focuses on spatial instabilities and patterns. We examine the simplest type of spatial pattern in standard reaction-diffusion systems in Chap. 9, namely patterns in a finite domain where the density vanishes at the boundaries. We discuss methods to determine the smallest domain size that supports a nontrivial steady state, known as the critical patch size in ecology. In Chap. 10, we provide first an overview of the Turing instability in standard reaction-diffusion systems. Then we explore how deviations from standard diffusion, namely transport with inertia and anomalous diffusion, affect the Turing instability. Chapter 11 deals with the effects of temporally or spatially varying diffusivities on the Turing instability in reaction-diffusion systems. We present applications of Turing systems to chemical reactions and biological systems in Chap. 12. Chapter 13 deals with spatial instabilities and patterns in spatially discrete systems, such as diffusively and photochemically coupled reactors. 

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. 

Sometimes very low Tafel slopes are claimed (<15mV) [99,253]. It seems difficult to interpret such an observation in terms of a specific mechanism. It is more probable that anomalously low Tafel slopes are the result of a combined thermal activation of the reaction and of the electrode surface state, resulting in a reaction rate limited by the diffusion of molecular hydrogen away from the electrode. Supersaturation of the electrode ad-layer by the evolved gas can also play a decisive role [254,255]. This phenomenon has been amply discussed in the case of Cl2 evolution on oxide electrodes [256], but the same idea can be applied to the case of H2 evolution [257,258],... 

The free energy required for this process is approximately 2.303RT (14-pKhb), and this is the cause of the activation energy barrier to the proton transfer. This conclusion is, of course, equally valid for excited and unexcited species, and is extremely unusual in that most workers consider such proton transfers to be concerted processes. The rates of these anomalous reactions are all so fast that they might be considered to be diffusion-controlled, and it is only the accuracy with which eqn. (53) predicts the rate of the other reactions that makes these reactions seem anomalous. 

Since translational diffusion process is sensitive to the microscopic structure in the solution, understanding the diffusion provides an important insight into the structure as well as the intermolecular interaction. Therefore, dynamics of molecules in solution have been one of the main topics in physical chemistry for a long time. 1 Recently we have studied the diffusion process of transient radicals in solution by the TG method aiming to understand the microscopic structure around the chemically active molecules. This kind of study will be also important in a view of chemical reaction because movement of radicals plays an essential role in the reactions. Here we present anomalous diffusion of the radicals created by the photoinduced hydrogen abstraction reaction. The origin of the anomality is discussed based on the measurments of the solvent, solute size, and temperature dependences. 

On the other hand, the effectiveness of the signaling reactions also depends on the diffusion coefficient, as shown in Eq. (33.11). Although other parameters in Eq. (33.11) (rs, rA, and nT) are not variable as determined for each reaction (33.10), only the diffusion coefficients (Ds and DA) can be controlled by the existence of the surrounding media. Moreover, as mentioned in the previous section, the diffusion coefficient of anomalous diffusion depends on the diffusion time and the dimensions of the reaction space. In such a situation, the diffusion coefficient observed by one method (e.g., FCS, FRAP) is only a local value, depending on the time constant and the spatial size of a proper experiment. As mentioned for Figure 33.4 in the beginning of this section, the size of the reaction volume for signaling reaction is of the order of pL-fL and measurement of the diffusion coefficient in such a microspace is important. 

Diffusion coefficients are typically higher in SCFs than in liquids. This is partly because the substances used as the solvent, such as carbon dioxide, have typically lighter and smaller molecules than organic liquid solvents and partly because the density of an SCF is typically less than a liquid. Consequently, reactions controlled by diffusion may be faster than in a liquid, giving the advantage of smaller process plant size. However, in the region of the critical point, diffusion coefficients can show an anomalous lowering, which can effect reaction rates. The behavior of diffusion coefficients is therefore discussed in Section 1.3.1 and its effect on reactions in Section 1.3.2. 

We plot in Fig. 6.5 the dimensionless front velocity vt/u vs the reaction rate r on a log-log scale. The front velocity increases with r. For the cases / = a and I = 2a, the slope is very similar, but for / = oo it is steeper. In all cases the front velocity increases as a power law of r, straight line in a log-log plot, for small and moderate values of r and saturates to 1 for larger values, the slope in the log-log plot tends to 0. This behavior is due to the fact that an increase of the reaction rate r leads to an increase of the front velocity. However, the front cannot travel faster than the jump velocity of the particles if all of them jump in the backbone direction, i.e., V < ajx. For I = a and / = 2a the transport is diffusive, and the diffusion coefficient is properly defined. If this transport is combined with a KPP reaction, a Fisher velocity is expected, i.e., in both cases v fr. Computing numerically the slope from a linear fit in Fig. 6.5 we obtain and for / = a and I = 2a, respectively. The case / oo is quite different, because the transport is anomalous. Equation (5.36) with y = 1/2 yields v while the linear fit of the numerical results yields Numerical and analytical results are in good agreement. 

Santamaria et al. [375] found that the transport of biologically inert particles, fluorescein dextran, in spiny dendrites is very slow compared with standard diffusion. The mean-square displacement is x t)) with y < 1 [298, 379]. The anomalous diffusion appears to be caused by the dendritic spines acting as the traps for the particles. We present here a mesoscopic model for the transport and biochemical reactions inside a population of spines and dendrites [122]. The morphology of spiny dendrites is very complex the distances between the spines and their sizes and shapes are randomly distributed [179, 362]. The model allows us to deal with the morphological diversity of dendritic spines via the transparent formalism of waiting time distributions. 

Yuste, S.B., Lindenberg, K. Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents. Phys. Rev. E 72(6), 061103 (2005). http //dx.doi.org/10.1103/PhysRevE.72.061103... 

One of such tendencies is polymers synthesis in the presence of all kinds of fillers, which serve simultaneously as reaction catalyst [26, 54]. The second tendency is the chemical reactions study within the framework of physical approaches [55-59], from which the fractal analysis obtained the largest application [36]. Within the framework of the last approach in synthesis process consideration such fundamental conceptions as the reaction prodrrcts stracture, characterized by their fractal (Hausdorff) dimension [60] and the reactionary medium connectivity, characterized by spectral (fracton) dimension J [61], were introduced. In its titrrt, diffusion processes for fractal reactions (strange or anomalous) differ principally from those occurring in Euclidean spaces and described by diffusion classical laws [62]. Therefore the authors [63] give transesterification model reaction kinetics description in 14 metal oxides presence within the framework of strange (anomalous) diffusion conception. 

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