Continuous-Time Random Walk

We now turn to a particle that performs a random walk in continuous time. In order to keep this section as clear as possible, we will only consider one-dimensional random walk models. As before, we assume that the jumps Zi, Z2. are independent identically distributed (IID) random variables. However, the jumps occur at random times Tj, T2,. .., so that the intervals between jumps are also HD 

In the mathematical literature, X (t ) is called a semi-Markov process associated with the two-component Markov chain X , T ), a Markov renewal process [218]. As discussed in Sect. 2.3, the CTRW model is a standard approach for studying anomalous diffusion [298]. 

The microscopic stochastic equation for the particle position X (t) can be written in the form 

3 Random Walks and Mesoscopic Reaction-Transport Equations 

Applying the Fourier-Laplace (F-L) transform to (3.28) and (3.30), we obtain the Fourier-Laplace transform of the PDF p(x, t), the Montroll-Weiss equation, 

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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... 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

In the continuous-time random walk model, a random walker is pictured to execute jumps at time steps chosen from the waiting time pdf w(t). In the isotropic and homogeneous (that is, force-free) case, the distance covered in a single jump event can be drawn from the jump length pdf X x). Then, the probability t) (x, t) of just having arrived at position x is given through [49]... 

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. 

How general are our results From a stochastic point of view ergodicity breaking, Levy statistics, anomalous diffusion, aging, and fractional calculus, are all related. In particular ergodicity breaking is found in other models with power-law distributions, related to the underlying stochastic model (the Levy walk). For example, the well known continuous time random walk model also... 

THE CONTINUOUS-TIME RANDOM WALK VERSUS THE GENERALIZED MASTER EQUATION... 

Equation (20) is the central result of the Zwanzig projection method, and it is one of the two theoretical tools under scrutiny in this chapter, the first being the Generalized Master Equation (GME), of which Eq. (20) is a remarkable example, and the second being the Continuous Time Random Walk (CTRW) [17]. It must be pointed out that to make Eq. (20) look like a master equation, it is necessary to make the third term on the right-hand side of it vanish. To do so, the easiest way is to make the following two assumptions ... 

Let us illustrate now the second of the two main theoretical tools under discussion in this review, this being the Continuous Time Random Walk (CTRW) proposed many years ago by Montroll and Weiss [17]. According to the CTRW we write... 

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