Normal random-walk diffusion

The statistics of the normal distribution can be applied to give more information about random-walk diffusion. The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of + J = + v/(2/V) on either side of it is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2 J, that is, 2V(2Dr t) is equal to about 5%. 

How does the ion move on the surface It cannot drift under an electric field because the field at an interface is normal to the electrode surface (Fig. 7.131) and what is under discussion here is motion parallel to the surface plane. The movements are by a random-walk diffusion process in two dimensions, surface diffusion. 

In contrast to normal diffusion, Ar2n does not grow linearly but with the square root of time. This may be considered the result of superimposing two random walks. The segment executes a random walk on the random walk given by the chain conformation. For the translational diffusion coefficient DR = kBT/ is obtained DR is inversely proportional to the number of friction-performing segments. 

The random-walk model of diffusion can also be applied to derive the shape of the penetration profile. A plot of the final position reached for each atom (provided the number of diffusing atoms, N, is large) can be approximated by a continuous function, the Gaussian or normal distribution curve2 with a form ... 

When the random-walk model is expanded to take into account the real structures of solids, it becomes apparent that diffusion in crystals is dependent upon point defect populations. To give a simple example, imagine a crystal such as that of a metal in which all of the atom sites are occupied. Inherently, diffusion from one normally occupied site to another would be impossible in such a crystal and a random walk cannot occur at all. However, diffusion can occur if a population of defects such as vacancies exists. In this case, atoms can jump from a normal site into a neighboring vacancy and so gradually move through the crystal. Movement of a diffusing atom into a vacant site corresponds to movement of the vacancy in the other direction (Fig. 5.7). In practice, it is often very convenient, in problems where vacancy diffusion occurs, to ignore atom movement and to focus attention upon the diffusion of the vacancies as if they were real particles. This process is therefore frequently referred to as vacancy diffusion... 

Fig. 7.131. Since the surface adion is unaffected by the electric field normal to the electrode surface, in order to reach the step site, the adion diffuses in a random-walk manner.

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

Consider a situation in an electrolytic solution where the concentration of the ionic species of interest is constant in the yz plane but varies in the x direction. To analyze the diffusion of ions, imagine a unit area of a reference plane normal to the x direction. This reference plane will be termed the transit plane of unit area (Fig. 4.14). There is a random walk of ions across this plane both from left to right and from right to left. On either side of the transit plane, one can imagine two planes L and R that are parallel to the transit plane and situated at a distance from it. In other words, Ihe region under consideration has been divided into left and right compartments in which the concentrations of ions are different and designated by and c, respectively. 

We can now show the relation between the random walk analogy (Fig. 9.1) and the diffusion coefficient. If we released a cluster of random walk particles with total mass Q, at x = 0 and t = 0, they would describe a normal distribution about x = 0 for t > 0. The number of particles, M, occupying a distance AI along the line is given by... 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

When we use a time step of constant interval At (t = i At), the diffusion process can be expressed as a simple sequence of A% . This random walk model is convenient to express the normal Brownian motion [9,10]. The definition of Brownian motion,... 

Figure 33.3 Different results of motion picture with different video frame speeds for the identical trajectory of random walk. In contrast to the case of normal diffusion, observed mean square displacements (MSD) significantly depend on the frame speed in the case of anomalous diffusion.

This concept, which is based on a random walk with a well-defined characteristic time and which applies when collisions are frequent but weak [13], leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 71, due to Fiirth), we obtain the Fokker Planck equation for the evolution of the distribution function in phase space which describes normal diffusion. 

Together Eqs (7.18) and (7.23) express the essential features of biased random walk A drift with speed v associated with the bias kr ki, and a spread with a diffusion coefficient D. The linear dependence of the spread (fe ) on time is a characteristic feature of normal diffusion. Note that for a random walk in an isotropic three-dimensional space the corresponding relationship is... 

Mazo (1998) studied Taylor dispersion in fractal media and found that the proportionality constant between the spatial spreading of a solute pulse and the time depended on both the fractal dimension of the medium and the dimension of the random walk through it. In normal diffusion the average particle position is directly proportional to the time. Diffusion in fractal media is anomalous with proportional to f2/dt, where dt is the random walk dimension. 

Figure 2.1 Normal diffusion in a two-dimensional random walk with...

Contrary to expectations, on the millisecond time scale the initial drop //< in the luminescence studies of Mays and Ilgenfritz [24] did not increase significantly with the temperature-induced cluster growth but remained constant even when an infinite percolation cluster was present. Furthermore, the observed decays were always exponential (Fig. 9). Evidently, the initial drop no longer reflects the cluster size. The process responsible for it is over within 50 //s and should perhaps rather be looked upon similarly to the transient active sphere part of normal diffusion-controlled decay. The diffusion in this case is a random walk performed by the quencher on a (percolation) cluster. A stretched exponential decay would be expected for a random walk deactivation on a static cluster, as was observed close to the percolation threshold in earlier studies [23,24]. Those measurements were performed over a time window of about 50 //s, which is close to the reported value of the cluster lifetime from electrical birefringence measurements [60]. It is very likely that... 

As far as transport properties of a fractal structure are concerned, the mean square displacement (MSD) of a particle follows a power law, (r ) where r is the distance from the origin of the random walk and is known as the random walk dimension. In other words, diffusion on fractals is anomalous, see Sect. 2.3. Recall that for normal diffusion in three-dimensional space the MSD is given by (r ) = 6Dt. For fractals, dy, > 2, and the exponent of t in the MSD is smaller than 1. We introduce a dimensionless distance by dividing r by the typical diffusive... 

Equation (6.12) faces one main difficulty, namely the determination of the explicit functional form of D(s). It is important to stress that D s) does not have exactly the same properties as the classical diffusion coefficient, and we refer to it here as the conductivity. Likewise, we define the resistivity of the medium as R = l/D. We expect the resistivity to be proportional to the number of steps of the particle, and arguments from random walks on fractals should be useful to determine R. Walks on fractals are characterized by the existence of two scales. Divide the medium into small blocks of size, such that the diffusion is normal within the small blocks, f. At scales larger than f, the effect of the heterogeneities becomes important, and motion depends on the fractal parameters. The self-invariance properties of the fractal are not valid at short distances. Similarly, the idealized concept of self-similarity at all scales does not hold for fractals in practice. 

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. 

Neither the electron density dependence nor the shape (which is approximately stretched exponential) of the kinetics can be explained with second order reaction kinetics, where it is assumed that the reaction is controlled only by the concentrations of electrons and dye cations, nor are they consistent with simple electron transfer theory. An explanation was proposed by Nelson based on the continuous time random walk [109]. In the CTRW, electrons perform a random walk on a lattice, which contains trap sites distributed in energy, according to some distribution function, g E). In contrast to normal diffusion, where the mean time taken for each step is a constant, in the CTRW the time taken for each electron to move is determined by the time for thermal escape from the site currently occupied. 

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