Diffusion and Random Walks

It should be noted that in one- or two-dimensional systems the mathematical expressions for diffusion and random walk are quite different from three-dimensional results This can be especially important in certain organic crystals and polymers. 

Consider now the process of diffusion in a thin layer of liquid, into which a substance (particles) with concentration Co is introduced at the initial moment t = 0, at the point x = 0. The substance diffuses in the liquid. In view of the analogy between diffusion and random walk of particles, it is assumed that in time t, the particle makes n displacements, where the number n is proportional to t ... 

The connection between diffusion and random walks was given by Einstein in a study of Brownian motion [34]. Suppose a collection of points is initially sampled fromp(.Y, t = 0) and that after a short time, each point takes an independent random step sampled from a three dimensional Gaussian distribution ... 

Interest in the Williams-Watts approach has arisen, not only because of its empirical success in fitting dielectric data, but also because of its relation to certain types of diffusion and random walk problems. The mechanistic relation between diffusion and relaxation was introduced by Glarum [1960], who suggested a process in which a mobile defect enabled a frozen in dipole to relax. Further aspects of random walk processes and their relation to CEPs and other empirical functions are discussed in a later section. 

There is an intimate connection at the molecular level between diffusion and random flight statistics. The diffusing particle, after all, is displaced by random collisions with the surrounding solvent molecules, travels a short distance, experiences another collision which changes its direction, and so on. Such a zigzagged path is called Brownian motion when observed microscopically, describes diffusion when considered in terms of net displacement, and defines a three-dimensional random walk in statistical language. Accordingly, we propose to describe the net displacement of the solute in, say, the x direction as the result of a r -step random walk, in which the number of steps is directly proportional to time ... 

Adatom diffusion, at least under the low temperature of field ion microscope measurements, almost always follows the direction of the surface channels. Thus adatoms on the W (112) and Rh (110) surfaces diffuse in one direction along the closely packed atomic rows of the surface channels. Such one-dimensional surface channel structures and random walks can be directly seen in the field ion images, and thus the diffusion anisotropy is observed directly through FIM images. Unfortunately, for smoother surfaces such as the W (110) and the fee (111), no atomic or surface channel structures can be seen in field ion images. But even in such cases, diffusion anisotropy can be established through a measurement of the two-dimensional displacement distributions, as discussed in the last section. Because of the anisotropy of a surface channel structure, the mean square displacements along any two directions will be different. In fact this is how diffusion anisotropy on the W (110) surface was initially found in an FIM observation.120... 

Two basic components of plate height in packed columns are Hf and HD, given by Eqs. 11.23 and 11.28, respectively. When these two terms are equal—which occurs at the specific velocity v c = AICm—we are at the transition point between a flow-controlled and diffusion-controlled random walk. (This transition, as it turns out, occurs near the minimum of each of the plate height curves shown in Figure 12.2.) The velocity u is a fundamental parameter characterizing every packed column system. Its value can be expressed by (see Eqs. 11.23 and 11.28)... 

Particles are transported along linear trajectories Particles adhere irreversibly to the aggregate upon contact N number of particles diffuse along random walk trajectories and aggregate upon collison Simulation ends with a single aggregate composed of N primary particles... 

The above-defined df and dt are structural parameters characterizing only the geometry of a given medium. However, when we are interested in processes like diffusion or reactions in disordered media, we need functional parameters, which are associated with the notion of time in order to characterize the dynamic behavior of the species in these media. The spectral or fracton dimension ds and random-walk dimension dw are two such parameters, and they will be defined in Section 2.2. 

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. 

In this regime the typical distance from the origin of motion increases as the square root of time. Thus, the dispersion in turbulent flows at long times is analogous to molecular diffusion or random walks with independent increments and comparison of Eq. (2.24) with (2.16) relates the turbulent diffusion coefficient, Dt, to the integral of the Lagrangian correlation function, Tl, as... 

The process of intersite electron hopping has been discussed in terms of a quasi-diffusional process. We now take a more detailed view of the intersite electron transfer reaction in a fixed-site redox polymer. The approach adopted here is due to Fritsch-Faules and Faulkner. These researchers developed a microscopic model to describe the electronhopping diffusion coefficient Z>e in a rigid three-dimensional polymer network as a function of the redox site concentration c. The model takes excluded volume effects into consideration, and it is based on a consideration of probability distributions and random-walk concepts. The microscopic approach was adopted by these researchers to obtain parameters that could be readily understood in the context of the polymer s molecular architecture. A previously published related approach was given by Feldberg. ... 

HC Berg, Random Walks in Biology. Princeton University Press, Princeton, 1993. Elementary and concise overview of diffusion, flow, and random walks. 

The very simple and somewhat ad hoc form of Glarum s assumptions coupled with the one dimensional diffusion model used have led to several extensions of the original treatment these include relaxation by next nearest and other neighboring defects again by diffusion or random walk models and to the three dimensioanl case by such models. These go well beyond the motivation of the original idea to see whether a simple cooperative mechanism could account for marked short time deviations from simple models of rotational relaxation of a single dipole in a mean field approximation. For a review of much of this see B ttcher-Bordevijk (48). Particularly in the three dimensional case however there is an increasing question as to how far one can or should invoke isotropic diffusion processes to relax the component of a molecular electric moment parallel to that at an earlier time. 

In a liquid, ion diffusion results from the random walk of ions, and their drift in an electric field is only superimposed on their random walk. Hence, the drift and random walk are linked. This is expressed by the Einstein relation between the mobility K and the diffusion coefficient D KU, with U = kT/e. 

In the presence of interactions between the connected segments of a single chain, aforementioned simple diffusion or random walks get affected and the walks are no more random. However, the intricate coupling of the different components such as monomers, solvent, or small ions in the case of polyelectrolytes via the interaction potentials complicates the theoretical analysis. In order to decouple different components, the conformations of the chain can be envisioned as the walks in the presence of fields, which arise solely due to the fact that there are interactions present in the system. This physical argument is the basis of the use ofcertain field theoretical transformations such as Hubbard-Stratonovich [60] transformation, which is well known in the field theory. So, the conformational characteristics of a polymer chain in the presence of different kinds of intrachain interactions can be described once the fields are known. In general, an exact computation of these fields is almost an impossible task. That is the reason theoretical developments resort to certain approximations for computing these fields, which work well for most of the practical purposes. Once these fields are known, the physical properties can be described in terms of these fields. It was shown by Edwards [50] that the similar analysis can be carried out for systems with many chains, where interchain interactions also affect the properties in addition to intrachain interactions. 

When we discussed random walk statistics in Chap. 1, we used n to represent the number of steps in the process and then identified this quantity as the number of repeat units in the polymer chain. We continue to reserve n as the symbol for the degree of polymerization, so the number of diffusion steps is represented by V in this section. 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

Equation (14), although derived from the approximate random walk theory, is rigorously correct and applies to heterogeneous surfaces containing wide variations in properties and to perfectly uniform surfaces. It can also be used as the starting point for the random walk treatment of diffusion controlled mass transfer similar to that which takes place in the stationary phase in GC and LC columns. 

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