Random walk approach

Later work on axial dispersion of particles has been carried out by Dorgelo et al. l44j who used an random-walk approach. 

Since the random-walk approach is successful in molecular diffusion (K5) and Brownian motion studies (C14), it would seem that it might also be useful for the dispersion process. This has been considered by Baron (B2), Ranz (Rl), Reran (B5), Scheidegger (S6), Latinen (L4) and more recently by de Josselin de Jong (D14) and Salfman (SI, S2, S3). The latter two did not strictly use random-walk since a completely random process was not assumed. Methods based on statistical mechanics have been proposed by Evans et al. (E7), Prager (P8), and Scheidegger (S7). 

In Sect. 2.1, the timescale over which the diffusion equation is not strictly valid was discussed. When using the molecular pair analysis with an expression for h(f) derived from a diffusion equation analysis or random walk approach, the same reservations must be borne in mind. These difficulties with the diffusion equation have been commented upon by Naqvi et al. [38], though their comments are largely within the framework of a random walk analysis and tend to miss the importance the solvent cage and velocity relaxation effects. 

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 ... 

Levy diffusion is a Markov process corresponding to the conditions established by the ordinary random walk approach with the random walker making jumps at regular time values. To explain why the GME, with the assumption of Eq. (112), yields Levy diffusion, we notice [50] that the waiting time distribution is converted into a transition probability n(x) through... 

A "microscopic probabilistic" method can be used for the modeling of linear chromatography. In this case, the probability density function at I and t of a single molecule of solute is derived. The "random walk" approach [29] is the simplest method of that type. It has been used to calculate the profile of the chromatographic band in a simple way, and to study the mechanism of band broadening. 

The random walk approach, which is based on Flory s treatment of the polymer chain in solution in this case the surface was considered as a reflecting barrier. 

The random walk approach is based on the random-walk concept, which was originally apphed to the problem of diffusion and later adopted by Flory [3] to deduce the conformations of macromolecules in solution. The earliest analysis was by Simha et al. [4], who neglected volume effects and treated the polymer as a random walk. Basically, the solution was represented by a three-dimensional lattice. 

Calculations of this type are particularly applicable to the flushing or rinsing of ion exchange columns with distilled water. In this case, D = 0. For beds with D = 0 and n < 20, Jacques and Vermeulen (Jl) have found that Einstein s random-walk approach (E2) gives a result which is intermediate to the diffusion equation (A4, M8) and the Poisson distribution (K6), and is thus most apt to apply to actual packed-bed conditions 4... 

Saffman (1959) developed a more general network model that accounted for interactions between molecular diffusion and fluid convection within individual tubes. Using the same random walk approach as de Josselin de Jongt I95K), he obtained numerical approximations of A for various limiting conditions, including... 

Berkowitz, B., and C. Braester. 1991. Dispersion in sub-representative elementary volume fracture networks Percolation theory and random walk approaches. Water Resour. Res 27 3159-3164. Berkowitz, B., and R.P. Ewing. 1998. Percolation theory and network modeling applications in soil physics. Surv. Geophys. 19 23-72. 

On the other hand, in the atomistic approach, the time-dependent configurations of the system are determined from the probabilities of the elementary atonuc process. The random walk approach calculates the probability of finding the system in a certain state after a certain time given an initial distribution of particles. It is then possible to show that the distributions are solutions of the diffusion equation. 

Let us recall the main assumptions of the simple model presented so far. (a) Whether we use the concentration-gradient or the random-walk approach, the reactant molecules are... 

A general transport phenomenon in the intercalation electrode with a fractal surface under the constraint of diffusion mixed with interfadal charge transfer has been modelled by using the kinetic Monte Carlo method based upon random walk approach (Lee Pyim, 2005). Go and Pyun (Go Pyun, 2007) reviewed anomalous diffusion towards and from fractal interface. They have explained both the diffusion-controlled and non-diffusion-controlled transfer processes. For the diffusion coupled with facile charge-transfer reaction the... 

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