Model random walk

2 MECHANISMS OF IONIC CONDUCTIVITY 6.2.1 Random-Walk Model 

INTRINSIC AND EXTRINSIC DEFECTS IN INSULATORS IONIC CONDUCTIVITY 

The number of jumps that an ion will make in the favored direction of the field per second can be analyzed as before (Section 5.7) to give  

For the field strengths used in conductivity experiments, ZeaEv is less than kT, so 

The mobility, p, of the ion is defined as the velocity when the value of Ev is unity, so  

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Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

At first glance it seems problematic whether anything is salvageable from the random walk model with so many areas of difference. 

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 (1) can be used in a general way to determine the variance resulting from the different dispersion processes that occur in an LC column. However, although the application of equation (1) to physical chemical processes may be simple, there is often a problem in identifying the average step and, sometimes, the total number of steps associated with the particular process being considered. To illustrate the use of the Random Walk model, equation (1) will be first applied to the problem of radial dispersion that occurs when a sample is placed on a packed LC column in the manner of Horne et al. [3]. 

This example of the use of the Random Walk model illustrates the procedure that must be followed to relate the variance of a random process to the step width and step frequency. The model will also be used to derive an expression for other dispersion processes that take place in a column. 

Dunn et al. (D7) measured axial dispersion in the gas phase in the system referred to in Section V,A,4, using helium as tracer. The data were correlated reasonably well by the random-walk model, and reproducibility was good, characterized by a mean deviation of 10%. The degree of axial mixing increases with both gas flow rate (from 300 to 1100 lb/ft2-hr) and liquid flow rate (from 0 to 11,000 lb/ft2-hr), the following empirical correlations being proposed ... 

This means that the precision of the prediction decreases with the square root of time. This describes the random walk model. A drift can be easily built into such a model by the addition of some constant drift function at each successive time period. 

A problem with this model is that very early values and recent values have an equal contribution to the precision of the forecast. The random walk model provides a good forecast of trend but is less efficient with cyclical and seasonal variations. 

Diffusion occurs when there is a concentration gradient of one kind of molecule within a fluid. In terms of random walk model, the average distance, x, after an elapsed time, t, between molecule collisions in a diffusion movement is characterized by the Einstein-Smoluchowski relation,... 

A second problem with the random walk model concerns the interaction between segments far apart along the contour of the chain but which are close together in space. This is the so-called "excluded volume" effect. The inclusion of this effect gives rise to an expansion of the chain, and in three-dimensions, 2 a, r3/5 (9), rather than the r dependence given in equation (I). 

For example, suppose a planar layer of N tracer atoms is the starting point, and suppose that each atom diffuses from the interface by a random walk in a direction perpendicular to the interface, in what is effectively one-dimensional diffusion. The probability of a jump to the right is taken to be equal to the probability of a jump to the left, and each is equal to 0.5. The random-walk model leads to the following result ... 

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... 

The random-walk model of diffusion needs to be modified if it is to accurately represent the mechanism of the diffusion. One important change regards the number of point defects present. It has already been pointed out that vacancy diffusion in, for example, a metal crystal cannot occur without an existing population of vacancies. Because of this the random-walk jump probability must be modified to take vacancy numbers into account. In this case, the probability that a vacancy is available to a diffusing atom can be approximated by the number of vacant sites present in the crystal, d], expressed as a fraction, that is... 

A second modification to the random-walk model of diffusion is required if motion is not random but correlated in some way with preceding passage through the crystal... 

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. 

Elements of the Transition Probability Matrix of a Unidimensional Unequiprobable Random Walk Model for an Ion Moving on an Electrode Surface 5i... 

Frasch HF (2002) A random walk model of skin permeation. Risk Anal 22 265-276. 

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