Random walk mean square displacement

The vacancy will follow a random-walk diffusion route, while the diffusion of the tracer by a vacancy diffusion mechanism will be constrained. When these processes are considered over many jumps, the mean square displacement of the tracer will be less than that of the vacancy, even though both have taken the same number of jumps. Therefore, it is expected that the observed diffusion coefficient of the tracer will be less than that of the vacancy. In these circumstances, the random-walk diffusion equations need to be modified for the tracer. This is done by ascribing a different probability to each of the various jumps that the tracer may make. The result is that the random-walk diffusion expression must be multiplied by a correlation factor, / which takes the diffusion mechanism into account. 

The correlation factor, for any mechanism, is given by the ratio of the values of the mean square displacement of the atom (often the tracer) moving in a correlated motion to that of the atom (or vacancy) moving by a random-walk process. If the number of jumps considered is large, the correlation factor/can be written as... 

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f . 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 (the root-mean-square displacement) 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 2f is equal to the total area under the curve minus the area under the curve up to 2f. This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

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) ... 

When aR > 1 and aL S> 1, we have p(il) p( l) for i > 1, and the walk becomes a nearest neighbor random walk. The mean displacement and mean square displacement are then given by... 

P(i, /, N) is the probability of finding the adatom, initially at site i, at site j after N jumps in an unrestricted random walk. The mean square displacement for a particular starting position, site i, is given by... 

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. 

Describe an experiment to determine Avogadro s number from the average root mean square displacement of a particle due to random walk. 

Some qualitative comments were made about the velocity autocorrelation function in Chap. 8, Sect. 2.1. In this section, it is considered in more quantitative detail. One of the simplest expressions for the diffusion coefficient is that due to Einstein [514]. He found that a particle executing a random walk has an average mean square displacement of (r2 > after a time t, such that... 

The random walk process can be characterized by the distribution of total displacements for either a large set of noninteracting walkers or for repeated trials of an isolated walker. The average displacement is a vector (R Nr))a.nd the mean-square displacement (R(NT) R(NT) = (R2(Nr)) is a scalar that characterizes the spread or diffuseness of the distribution of total displacements about its average. 

Equation 7.46 demonstrates that if each jump of a walk occurs randomly (i.e., is uncorrelated), the average displacement is zero and the center of mass of a large number of individual random jumpers is not displaced. Equation 7.47 gives the mean-square displacement of a random walk, NT(r2). Although Eqs. 7.46 and 7.47 were derived here for one-dimensional random walks, both are valid for two- and three-dimensional random walks. 

The probability distribution of a random walk shows that the mean-square displacement after NT jumps is (R2) = NT r2) = rY(r2) (Eq. 7.47). Comparison of the probability distribution (Eq. 7.45) to the point-source solution for one-dimensional diffusion from a point source (Table 5.1) indicates that... 

The calculated root-mean-square displacement for a general sequence of jumps has two terms in Eq. 7.31. The first term, NT(r2), corresponds to an ideal random walk (see Eq. 7.47) and the second term arises from possible correlation effects when successive jumps do not occur completely at random. 

In the previous section we analyzed the random walk of molecules in Euclidean space and found that their mean square displacement is proportional to time, (2.5). Interestingly, this important finding is not true when diffusion is studied in fractals and disordered media. The difference arises from the fact that the nearest-neighbor sites visited by the walker are equivalent in spaces with integer dimensions but are not equivalent in fractals and disordered media. In these media the mean correlations between different steps (UjUk) are not equal to zero, in contrast to what happens in Euclidean space cf. derivation of (2.6). In reality, the anisotropic structure of fractals and disordered media makes the value of each of the correlations u-jui structurally and temporally dependent. In other words, the value of each pair u-ju-i-- depends on where the walker is at the successive times j and k, and the Brownian path on a fractal may be a fractal of a fractal [9]. Since the correlations u.juk do not average out, the final important result is (UjUk) / 0, which is the underlying cause of anomalous diffusion. In reality, the mean square displacement does not increase linearly with time in anomalous diffusion and (2.5) is no longer exact. 

The type of the random walk (recurrent or nonrecurrent) determines the minimum value of the two terms in the brackets of the previous equation. If the walker does not visit the same sites (nonrecurrent) then dw = 2df/ds. If the walk is of recurrent type then the walker visits the same sites again and again and therefore the walker covers the available space (space-filling walk). Consequently, the meaning of dw coincides with df (dw = df). The mean square displacement in anomalous diffusion follows the pattern... 

As mentioned above, the diffusion process is thought to be a random walk across the surface. Then the mean-square displacement of the adparticles is related to the diffusion coefficient via the relation... 

In the fractal porous medium, the diffusion is anomalous because the molecules are considerably hindered in their movements, cf. e.g., Andrade et al., 1997. For example, Knudsen diffusion depends on the size of the molecule and on the adsorption fractal dimension of the catalyst surface. One way to study the anomalous diffusion is the random walk approach (Coppens and Malek, 2003). The mean square displacement of the random walker (R2) is not proportional to the diffusion time t, but rather scales in an anomalous way ... 

Note that in Eq. 6.34 the mean-square displacement is used, rather than the root-mean-square displacement. For a one-dimensional random walk, the mean-square displacement is given by 2Dt, and for a two-dimensional random walk, 4Dt. Since the jump distance (a vector) is A, if the jump frequency is now defined as F = n/t (the average number of jumps per unit time), then on combining Eq. 6.33 and 6.34 gives ... 

We remark that if the jump length distance is also a Levy process, the mean-square displacement does not exist which has led to conceptual difficulties in applying this process to dielectric relaxation. Using these simplifications, one can identify two specialized forms of a continuous time random walk ... 

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