Random walk model of diffusion

As there is equal probability of -t-/ and -I, the mean value of x N) is equal to zero. Hence, x N) does not reveal the diffusion kinematics. Standard deviation of a quantity is the measure of data spread out defined as  

Transport Phenomena in Microfluidic Systems, First Edition. Pradipta Kumar Panigrahi. 

The square of standard deviation of the final particle position can also be written as 

The diffusion length for Af-steps in 1 -D can be defined as square root of the standard deviation given as 

The total time t elapsed after N equal time steps of t is given by 

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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 ... 

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... 

Thomson, D. J. (1984). Random walk modelling of diffusion in inhomogeneous turbulence. Quart. J. Roy. Meteorol. Soc. 110,1108-1120. 

Figure 33.2 Two vector models expressing the random walk (trajectory) of diffusion, (a) Sequence of position vectors on each...

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

Diffusion results from Brownian motion, the random battering of a molecule by the solvent. Let s apply the one-dimensional random walk model of Chapter 4 (called random flight, in three dimensions) to see how far a peirticle is moved by Brownian motion in a time t. A molecule starts at position x = 0 at time t = 0. At each time step, assume that the particle randomly steps either one unit in the +x direction or one unit in the -x direction. Equation (4.34) gives the distribution of probabilities (which we interchangeably express as a concentration) c(x, N) that the particle will be at position x after N steps,... 

The connection of random-walk calculations based on concentration-gradient theory may be summoned as follows. As we have noted (Section 2.3), the Equation for diffusion-controlled rate constants based on Pick s law agrees with that derived from a random-walk model (Section 2.3). Pick s law is in fact a macroscopic consequence of the random-walk model of molecular-scale processes or, to put it the other way round, the random-walk model is an interpretation of Pick s experimental law [4]. 

Random-walk models for diffusion on the lattice can be implemented in various ways. Typically, particles randomly hop between a node r and its neighbors r e -T"(r). If there are no restrictions on the number of particles at a node, the stationary distribution for this diffusive stochastic process is Poissonian, ... 

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. 

CA in which many filled cells execute a random walk but never interact with one another, cannot give rise to stable pattern formation since the cells will move at random forever. However, if cells can interact when they meet, so that one diffusing cell is allowed to stick to another, stable structures can be created. These structures illustrate the modeling of diffusion-limited aggregation (DLA), which is of interest in studies of crystal formation, precipitation, and the electrochemical formation of solids. 

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,... 

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... 

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,... 

Below we will show that diffusivity D can be interpreted in the framework of a random walk model (see Eqs. 18-16 and 18-17). Particularly, D is related to the random walk parameters, mean free path X and mean velocity ux, by the simple relation ... 

The random walk model is certainly less suitable for a liquid than for a gas. The rather large densities of fluids inhibit the Brownian motion of the molecules. In water, molecules move less in a go-hit-go mode but more by experiencing continuously varying forces acting upon them. From a macroscopic viewpoint, these forces are reflected in the viscosity of the liquid. Thus we expect to find a relationship between viscosity and diffusivity. 

The coefficient Ex is called the turbulent (or eddy) diffusion coefficient it has the same dimension as the molecular diffusion coefficient [L2 1]. The index x indicates the coordinate axis along which the transport occurs. Note that the turbulentjliffusion coefficient can be interpreted as the product of a mean transport distance Lx times a mean velocity v = (Aa At) l Egex, as found in the random walk model, Eq. 18-7. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

The simplest of these models which permits a detailed discussion of the decay of correlations is a random walk model in which a set of random walkers whose positions are initially correlated is allowed to diffuse the motion of any single random walker being independent of any other member of the set. Let us assume that there are r particles in the set and motion occurs on a discrete lattice. The state of the system is, therefore, completely specified by the probabilities Pr(nlf n2,..., nr /), (tij = — 1, 0, 1, 2,. ..) in which Pr(n t) is the joint probability that particle 1 is at n1( particle 2 is at n2, etc., at time l. We will also use the notation Nj(t) for the random variable that is the position of random walker j at time t. Reduced probability distributions can be defined in terms of the Pr(n t) by summation. We will use the notation P nh, rth,..., ntj I) to denote the distribution of random walkers iu i2,..., i at time t. We define... 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

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