Random walk, .

For simplicity we describe first the one-dimensional random walk (see Chandrasekhar, 1943). Consider n steps of random walk along an axis, with each step either in the forward or the backward direction. Each step has equal probability 

The two extremes are all n steps directed to the right (positive) or all n steps directed to the left (negative) from the origin 0 along one dimension (x axis). Most probably n steps end in between n and —n since the walk is random. Let n+ be the positive step, n be the negative step, and m be the last step. Then m is the distance from the origin and m = n+ — ot m = ti- — n+. If n+ = n, m = 0. We take 

The probability of leading to the value of m after n steps of random walk can be expressed in terms of w n,m), which is the Bernoulli probability (or the Bernoulli distribution). Thus, for the random-walk distribution (probability distribution) 

This is exactly the Gaussian distribution function. Instead of m, we now introduce the net displacement r from the starting point as the variable that is, we change the 

Here Ax represents the intervals along the straight line and v(n,x) Ax is the probability that the random walk ends in the interval between x and x + Ax after n steps. Thus, we conclude that the random walk is governed by a distribution function v(/i,x) which is Gaussian in nature. 

We derive the probability for the position of a particle for the random walk [4]. We follow the position of a particle that moves in a linear spatial lattice with a mesh size of Ax. We observe the particle after uniform time steps. The particle may move in a time step with the probability p(x) at one grid point forward and move with the probability q(x) one grid point backward. Therefore, 

The particle will not rest at the same position during a time step. It will move either forward or backward. 

If one time step lasts At then n time steps will last t = nAt. This is what we have made use of in Eq. (21.5). We set now 

Otherwise, the variance will be in the limit for At 0 with the condition 

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Adsorbed atoms and molecules can also diflfiise across terraces from one adsorption site to another [33]. On a perfect terrace, adatom diflfiision could be considered as a random walk between adsorption sites, with a diflfiisivity that depends on the barrier height between neighbouring sites and the surface temperature [29]. The diflfiision of adsorbates has been studied with FIM [14], STM [34, 35] and laser-mduced themial desorption [36]. 

Viswanathan R, Raff L M and Thompson D L 1984 Monte Carlo random walk calculations of unimolecular dissociation of methane J. Chem. Phys. 81 3118-21... 

In the limit that the number of effective particles along the polymer diverges but the contour length and chain dimensions are held constant, one obtains the Edwards model of a polymer solution [9, 30]. Polymers are represented by random walks that interact via zero-ranged binary interactions of strength v. The partition frmction of an isolated chain is given by... 

Zumofen G and Klafter J 1994 Spectral random walk of a single molecule Chem. Phys. Lett. 219 303-9... 

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... 

In a random walk on a square lattice the chain can cross itself. 

As a consequence of these various possible conformations, the polymer chains exist as coils with spherical symmetry. Our eventual goal is to describe these three-dimensional structures, although some preliminary considerations must be taken up first. Accordingly, we begin by discussing a statistical exercise called a one-dimensional random walk. 

For a one-dimensional random walk, the probability of n j heads after n moves is supplied by application of the bionomial distribution formula ... 

This result enables us to calculate the probability of any specified outcome for the one-dimensional random walk. We shall continue to develop this one-dimensional relationship somewhat further, since doing so will produce some useful results. 

Next let us apply random walk statistics to three-dimensional chains. We begin by assuming isolated polymer molecules which consist of perfectly flexible chains. 

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... 

With this probability expression, it is an easy matter to calculate the average dimensions of a coil. Because of the back-and-forth character of the x, y, and z components of the random walk, the average end-to-end distance is less meaningful than the average of r. The latter squares positive and negative components before averaging and gives a more realistic parameter to characterize the coil. To calculate r, we remember Eq. (1.11) and write... 

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

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 ... 

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. 

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). 

Fig. 22.4. The random walk of o chain in a polymer melt, or in a solid, glassy polymer means that, on average, one end of the molecule is -yJn)A away from the other end. Very large strains (=4) are needed to straighten the molecule out.

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