Reaction random walk

Lattice models and other discrete models are used to describe a wide variety of dynamical systems [270, 77]. In this section we study the propagation of reaction-random walk wavefronts on heterogeneous lattices that consist of a main backbone with a regular distribution of secondary branches. An example is comb-like structures, see Fig. 6.3 [53]. 

As a third approach, we use persistent random walks with reactions, see Sect. 2.2.3, to study the effect of inertia on Turing instabilities [205, 206]. This is our preferred approach to describe reaction-transport systems with inertia, since it has a solid mesoscopic foundation. We consider Turing instabilities for two classes of reaction random walks, DIRWs and DDRWs, see Sect. 5.6. 

Horsthemke, W. Fisher waves in reaction random walks. Phys. Lett. A 263(4-6), 285-292... 

Horsthemke, W. Spatial instabilities in reaction random walks with direction-independent kinetics. Phys. Rev. E60(3),2651-2663 (1999). http //dx.doi.org/10.1103/PhysRevE. 60.2651... 

A molecular model for translational diffusion and diffusion-influenced rates of reaction random-walk theory... 

In this relatively simple random walk model an ion (e.g., a cation) can move freely between two adjacent active centres on an electrode (e.g., cathode) with an equal probability A. The centres are separated by L characteristic length units. When the ion arrives at one of the centres, it will react (e.g., undergoes a cathodic reaction) and the random walk is terminated. The centres are, therefore absorbing states. For the sake of illustration, L = 4 is postulated, i.e., Si and s5 are the absorbing states, if 1 and 5 denote the positions of the active centres on the surface, and s2, s3, and s4 are intermediate states, or ion positions, LIA characteristic units apart. The transitional probabilities (n) = Pr[i-, —>, Sj in n steps] must add up to unity, but their individual values can be any number on the [0, 1] domain. 

In the simplest case, the random walk is equi-probable between two adjacent positions, but no jumpover is allowed, i.e., P3i = Pai= =0 and the ion cannot be stagnant, i.e., p22 = Pa = P44 = 0 (however, pu = Pa = 1). Table 1 indicates that the ion eventually reacts at one of the active centres, the probability of reaction at a centre depending on the initial position of the ion. If, e.g., the initial position was, s4, the probability of the ion arriving eventually at the active centre s is %, and at active centre s5, 3/i The final state is reached essentially after about 25 steps (stages). 

Ramsey theory, 22 201-204 Random-fragmentation model, Szilard-Chalmers reaction and, 1 270 Random-walk process, correlated pair recombination, post-recoil annealing effects and, 1 288-290 Rare-earth carbides, neutron diffraction studies on, 8 234-236 Rare-earth ions energy transfer, 35 383 hydration shell, 34 212-213 Rare gases... 

To keep the reaction going and the electronation current constant, a steady supply of electron-acceptor ions must be maintained by transport from the electrolyte bulk. This transport may be by diffusion (random walk) or migration under an electric field (drift) [cf. Eq. (4.226)]. 

It is convenient to label the relative slowness of encounter pair reaction as due to an activated process and to remark that the chemical reaction (proton, electron or energy transfer, bond fission or formation) can be activation-limited. This is an unsatisfactory nomenclature for several reasons. Diffusion of molecules in solution not only involves a random walk, but oscillations of the molecules in solvent cages. Between each solvent cage in which the molecule oscillates, a transformation from one state to another occurs by passage over an activation barrier. Indeed, diffusion is activated (see Sect. 6.9), with a typical activation energy 8—12 kJ mol-1. By contrast, the chemical reaction of a pair of radicals is often not activated (Pilling [35]), or rather the entropy of activation... 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

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