Continuous random walk

The simplest model of translational motions in three dimensions is the continuous random walk diffusion (Pick s law) giving a Lorentzian-shaped scattering law,... 

In RANS-based simulations, the focus is on the average fluid flow as the complete spectrum of turbulent eddies is modeled and remains unresolved. When nevertheless the turbulent motion of the particles is of interest, this can only be estimated by invoking a stochastic tracking method mimicking the instantaneous turbulent velocity fluctuations. Various particle dispersion models are available, such as discrete random walk models (among which the eddy lifetime or eddy interaction model) and continuous random walk models usually based on the Langevin equation (see, e.g.. Decker and... 

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. 

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. 

In this random walk analog with steps of fixed magnitude, m is permitted to assume only every other integral value hence x must change in steps of 2l/ /. To establish the connection with the distribution W(x) for the molecule, which is continuous in x, we note that ir(m,n) must equal TF(x)Ax, where Ax corresponds to Am —2. Hence... 

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 ... 

To account for the effect of a sufficiently broad, statistical distribution of heterogeneities on the overall transport, we can consider a probabilistic approach that will generate a probability density function in space (5) and time (t), /(i, t), describing key features of the transport. The effects of multiscale heterogeneities on contaminant transport patterns are significant, and consideration only of the mean transport behavior, such as the spatial moments of the concentration distribution, is not sufficient. The continuous time random walk (CTRW) approach is a physically based method that has been advanced recently as an effective means to quantify contaminant transport. The interested reader is referred to a detailed review of this approach (Berkowitz et al. 2006). 

Berkowitz B, Emmanuel S, Scher H (2008) Non-Fickian transport and multiple rate mass transfer in porous media Water Resour Res 44, D01 10.1029/2007WR005906 Bijeljic B, Blunt MJ (2006) Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 42, W01202, D01 10.1029/2005WR004578 Blunt MJ (2000) An empirical model for three-phase relative permeability. SPE Journal 5 435-445... 

In discussing the random walk and diffusion in Chapter 2, we use continuous functions as the weighting factors for calculating averages (e.g., Equation (2.63)). In the following section we discuss in greater detail the use of continuous weighting functions in statistics. 

To show the relationship between pn(m) expressing the probabilities of numbers and p x) describing a continuous spatial distribution of a quantity like concentration, we make use of the analogy between the integers n and m, which describe the simple random walk model shown in Fig. 18.1, and the time and space coordinates t and x, that is t = n At and x = m Ax. The incremental quantities, At and Ax, are characteristic for random motions the latter is the mean free path which is commonly denoted as X = Ax, the former is associated with the mean velocity ux= Ax/At = XIAt. Thus, we get the following substitution rules ... 

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. 

In order to calculate the density of reactant B about A, it is necessary to know by what means the reactants migrate in solution. Under most circumstances, diffusion is a very adequate description (the limitations of and complications to diffusion are discussed in Sect. 6, Chap. 8 Sect. 2 and Chap. 11). In this simple analysis of diffusion, Fick s laws will be used with little further justification, save to note that Fick s second law is identical to the equation satisfied by a random walk function. Hardly a surprising result, because diffusion is a random walk with no retention of information about where the diffusing species was before its current location. In Chap. 3 Sect. 1, the diffusion equation is derived from thermodynamic considerations and the continuity equation (law of conservation of mass). 

Exercise. The random walk with continuous time is defined as follows. The states are all integers n (— oo < n < oo). A particle can jump between neighboring states. In a short time dt has probability dt to jump to the right, and the same probability to jump to the left. Construct the master equation for pn(t) (compare VI.2). 

Exercise. An example with an infinite set of states to which the analysis of this section does not apply is the infinite random walk with continuous time... 

A fundamental property of the master equation is As t -> oo all solutions tend to the stationary solution or - in the case of decomposable or splitting W - to one of the stationary solutions. Again this statement is strictly true only for a finite number of discrete states. For an infinite number of states, and a fortiori for a continuous state space, there are exceptions, e.g., the random walk (2.11). Yet it is a useful rule of thumb for a physicist who knows that many systems tend to equilibrium. We shall therefore not attempt to give a general proof covering all possible cases, but restrict ourselves to a finite state space. There exist several ways of proving the theorem. Of course, they all rely on the property (2.5), which defines the class of W-matrices. 

This random walk differs from those considered in 1.4 and IV.5 in that the time varies continuously. The transition probabilities are now taken per unit time. This simple example often suffices to illustrate more complicated... 

Example. Consider the continuous-time, symmetric random walk... 

Exercise. For the random walk on a two-dimensional square lattice, either with discrete or continuous time, show that every lattice point is reached with probability 1, but on the average after an infinite time. In three dimensions, however, the probability of reaching a given site is less than unity there is a positive probability for disappearing into infinity. 

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