Directed random walk

The one-dimensional random walk of the last section is readily adapted to this problem once we recognize the following connection. As before, we imagine that one end of the chain is anchored at the origin of a three-dimensional coordinate system. Our interest is in knowing, on the average, what will be the distance of the other end of the chain from this origin. A moment s reflection will convince us that the x, y, and z directions are all equally probable as far as the perfectly flexible chain is concerned. Therefore one-third of the repeat units will be associated with each of the three perpendicular directions... 

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 ... 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

The orientation of linear rotators in space is defined by a single vector directed along a molecular axis. The orientation of this vector and the angular momentum may be specified within the limits set by the uncertainty relation. In a rarefied gas angular momentum is well conserved at least during the free path. In a dense liquid it is a molecule s orientation that is kept fixed to a first approximation. Since collisions in dense gas and liquid change the direction and rate of rotation too often, the rotation turns into a process of small random walks of the molecular axis. Consequently, reorientation of molecules in a liquid may be considered as diffusion of the symmetry axis in angular space, as was first done by Debye [1],... 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

Direct observation of molecular diffusion is the most powerful approach to evaluate the bilayer fluidity and molecular diffusivity. Recent advances in optics and CCD devices enable us to detect and track the diffusive motion of a single molecule with an optical microscope. Usually, a fluorescent dye, gold nanoparticle, or fluorescent microsphere is used to label the target molecule in order to visualize it in the microscope [31-33]. By tracking the diffusive motion of the labeled-molecule in an artificial lipid bilayer, random Brownian motion was clearly observed (Figure 13.3) [31]. As already mentioned, the artificial lipid bilayer can be treated as a two-dimensional fluid. Thus, an analysis for a two-dimensional random walk can be applied. Each trajectory observed on the microscope is then numerically analyzed by a simple relationship between the displacement, r, and time interval, T,... 

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 ... 

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... 

In the case of interstitial diffusion in which we have only a few diffusing interstitial atoms and many available empty interstitial sites, random-walk equations would be accurate, and a correlation factor of 1.0 would be expected. This will be so whether the interstitial is a native atom or a tracer atom. When tracer diffusion by a colinear intersticialcy mechanism is considered, this will not be true and the situation is analogous to that of vacancy diffusion. Consider a tracer atom in an interstitial position (Fig. 5.18a). An initial jump can be in any random direction in the structure. Suppose that the jump shown in Figure 5.18b occurs, leading to the situation in Figure 5.18c. The most likely next jump of the tracer, which must be back to an interstitial site, will be a return jump (Fig. 5.18c/). Once again the diffusion of the interstitial is different from that of a completely random walk, and once again a correlation factor, / is needed to compare the two situations. 

Translational motion is the change in location of the entire molecule in three-dimensional space. Figure 11 illustrates the translational motion of a few water molecules. Translational motion is also referred to as self-diffusion or Brownian motion. Translational diffusion of a molecule can be described by a random walk, in which x is the net distance traveled by the molecule in time At (Figure 12). The mean-square displacement (x2) covered by a molecule in a given direction follows the Einstein-derived relationship (Eisenberg and Crothers, 1979) ... 

We appropriate the random chain and notify as nt the numbers of steps of the end of chain random walk along /-directions of d-dimensional lattice. At this,... 

The random-walk model consists of a series of steplike movements for each molecule which may be positive or negative the direction being completely random. After (p) steps, each step having a length (s) the average of the molecules will have moved some distance from the starting position and will form a Gaussian type distribution curve with a variance of o2. ... 

Ordinary diffusion is the result of random molecular movement in first one direction and then another and thus, resembles the Random Walk Model. Uhlenbeck and Ornstein (8), derived the following expression for the overall standard deviation (o) arising from diffusion process,... 

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... 

Atomic jumps in random walk diffusion of closely bound atomic clusters on the W (110) surface cannot be seen. A diatomic cluster always lines up in either one of the two (111) surface channel directions. But even in such cases, theoretical models of the atomic jumps can be proposed and can be compared with experimental results. For diffusion of diatomic clusters on the W (110) surface, a two-jump mechanism has been proposed by Bassett151 and by Cowan.152 Experimental studies are reported by Bassett and by Tsong Casanova.153 Bassett measured the probability of cluster orientation changes as a function of the mean square displacement, and compared the data with those derived with a Monte Carlo simulation based on the two-jump mechanism. The two results agree well only for very small displacements. Tsong Casanova, on the other hand, measured two-dimensional displacement distributions. They also introduced a correlation factor for these two atomic jumps, which resulted in an excellent agreement between their experimental and simulated results. We now discuss briefly this latter study. 

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