Stable Distributions and Random Walks

Fluid flow and solute transport simulations in tight geologic formations: Discrete fracture network and continuous time random walk analyses

In recent years a huge interdisciplinary field has emerged which is devoted to the complex dynamics of anomalous transport with long-time memory and non-markovian features. It was found that the framework of fractional calculus and its generalizations are able to capture these phenomena. Many of the classical models are based on continuous-time renewal processes and use the Montroll Weiss continuous time random walk (CTRW) approach. On the other hand their discrete time counterparts are rarely considered in the literature despite their importance in various applications. The goal of the present paper is to give a brief sketch of our recently introduced discrete-time Prabhakar generalization of the fractional Poisson process and the related discrete-time random walk (DTRW) model. We show that this counting process is connected with the continuous time Prabhakar renewal process by a (well scaled) continuous-time limit. We deduce the state probabilities and discrete time generalized fractional Kolmogorov-Feller equations governing the Prabhakar DTRW and discuss effects such as long time memory (nonmarkovianity) as a hallmark of the complexity of the process.

We consider in this paper two types of queueing systems which operate under a specific and fixed scheduling policy. The first system consists of a single server and several buffers in which arriving jobs are stored. We assume that arriving parts may require several stages of processing in which case each stage corresponds to a different buffer. The second system is a communication type queueing network given by a graph. The arriving jobs (packets) request a simple path along which they need to be processed. In both models the jobs arrive in a completely deterministic fashion: the interarrival times are fixed and known. All the processing times are also deterministic. A scheduling policy specifies a rule using which arriving parts are processed in the queueing system. A scheduling policy is defined to be stable if there is a finite uniform upper bound on the total number of parts in the system at all times. A necessary condition for stability of any work conserving policy is that the traffic intensity of the station (of each link in the graph in the communication model) is not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([24],[20],[5],[6],[10]), deterministic fluid networks ([6],[9],[7],[8],[2]), deterministic adversarial networks ([4],[1],[14],[12]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queueing networks. Bramson [5] and Seidman [25] showed that even FIFO policy can be unstable in stochastic networks. Instability of FIFO was later demonstrated in an adversarial queueing setting by Andrews et. al. [1]. Dai [6] established that stability of a fluid deterministic queueing network implies stability of a stochastic queueing network. A similar result was established by Gamarnik [12], which connects stability of fluid and adversarial queueing networks. A complete characterization of two-station fluid networks which are stable under any work conserving policy was established by Bertsimas, Gamarnik and Tsitsiklis [2] and Dai and Vande Vate [7]. Goel [14] constructed a complete characterization of adversarial queueing networks which are stable under the usual load condition. The result is extended by Gamarnik [13]. Motivated by a queueing network model stability of homogeneous random walks in a nonnegative orthant was considered in several papers: Malyshev [21], Menshikov [23], Fayolle [11], Ignatyuk and Malyshev [18], Malyshev [22]. Such random walks Q ( t ), t = 0, 1, 2,¡­ have Z + d as a state space ( Z + is the set of nonnegative integers). The transition vectors ¦¤ have deterministically bounded length in max norm and the transition probability p (¦«, ¦¤) along the direction ¦¤ depends only on the face ¦« that the random walk is currently on (the transition probabilities depend only on which coordinates of the current state are positive and which are zero). Such a random walk is defined to be stable if it is positive recurrent. We will also consider deterministic walks, for which p (¦«, ¦¤) is always zero or one (the transition deterministically depends on the face that the walk is currently on). A complete characterization of stable homogeneous random walks in Z + d for d ¡Ü 4 was obtained in Malyshev [21], Menshikov [23] and Ignatyuk and Malyshev [18], but no extension of this classification to higher dimensions has been obtained. Malyshev in [22] establishes a connection between homogeneous random walks and general dynamical systems on compact manifolds and shows that the difficulty of classifying stable random walks is of the same nature as the difficulty of understanding the dynamics of these dynamical systems. Specifically, the complicated dynamics precludes obtaining classification of stable random walks for d = 5. Thus despite many efforts the classification of stable random walks for general dimensions is not known. Likewise classification of stable policies in queueing systems is an open problem.

In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

Continuous time random walks for non-local radial solute transport

Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Lévy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit.

Fickian and non-Fickian behaviors were often detected for contaminant transport activity owed to the preferential flow and heterogeneity of soil media. Therefore, using diverse methods to measure such composite solute transport in soil media has become an important research topic for solute transport modeling in soil media. In this article, the continuous-time random walk (CTRW) model was applied to illustrate the relative concentration of transport in low-permeability homogeneous and saturated soil media. The solute transport development was also demonstrated with the convection-dispersion equation (CDE) and Two Region Model (TRM) for comparison. CXTFIT 2.1 software was used for CDE and TRM, and CTRW Matlab Toolbox v.3.1 for the CTRW simulation of the breakthrough curve. It was found that higher values of determination coefficient (R2) and lower values of root mean square error (RMSE) concerning the best fits of CDE, TRM, and CTRW. It was found that in the comparison of CDE, TRM, and CTRW, we tend to use CTRW to describe the transport behavior well because there are prevailing Fickian and non-Fickian transport. The CTRW gives better fitting results to the breakthrough curves (BTCs) when β has an increasing pattern towards 2.00. In this study, the variation of parameters in three methods was investigated and results showed that the CTRW modeling approach is more effective to determine non-reactive contaminants concentration in low-permeability soil media at small depths.

The controversy about the precise form of the waiting time distribution for the first observed hop in the Scher and Lax description (1973) of AC hopping conductivity of disordered systems by a continuous time random walk (CTRW) on a lattice is resolved. Starting from an exact general definition of the above waiting time distribution for random walks in the actual disordered medium, the corresponding first hop waiting time distribution for the CTRW on a lattice is shown to coincide with the distribution of waiting times between consecutive hops, as assumed originally by Scher and Lax. The authors' analysis thus supports the correctness of the Scher and Lax theory of AC conductivity of disordered systems. The relation between the random walk equation for hopping in a random medium proposed by Scher and Lax and the more familiar rate equations for site-occupancy probabilities is also established.

Here we obtain almost sure limit points for a properly normalized partial sum continuous time random walk, where a continuous time random walk means a random walk subordinated to a renewal process. Continuous time random walks are used in physics to model anomalous diffusion.

Continuous-time random walks (CTRWs) are generic models of anomalous diffusion and fractional dynamics in statistical physics. They are typically defined in the way that their trajectories are discontinuous step functions. In this paper, we propose alternative definition of CTRWs with continuous trajectories. We also give the scaling limit theorem for sequence of such random walks. In general case this result requires the use of strong Skorohod [Formula: see text] topology instead of Skorohod [Formula: see text] topology, which is usually used in limit theorems for ordinary CTRW processes. We also give additional conditions under which convergence of sequence of considered random walks holds in the [Formula: see text] topology.

The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions psi(t) and general jump distribution functions eta(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times psi approximately t(-(1+beta)) and algebraic decaying jump distributions eta approximately x(-(1+alpha)) corresponding to Lévy stable processes, the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Lévy stochastic processes in the Lévy-Khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated Lévy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as tauc approximately lambda(-alpha/beta), where 1/lambda is the truncation length scale. The asymptotic behavior of the propagator (Green's function) of the truncated fractional equation exhibits a transition from algebraic decay for t<<tauc to stretched Gaussian decay for t>>tauc.

In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

Random walks have been proven to be useful for constructing various algorithms to gain information on networks. Algorithm node2vec employs biased random walks to realize embeddings of nodes into low-dimensional spaces, which can then be used for tasks such as multi-label classification and link prediction. The performance of the node2vec algorithm in these applications is considered to depend on properties of random walks that the algorithm uses. In the present study, we theoretically and numerically analyse random walks used by the node2vec. Those random walks are second-order Markov chains. We exploit the mapping of its transition rule to a transition probability matrix among directed edges to analyse the stationary probability, relaxation times in terms of the spectral gap of the transition probability matrix, and coalescence time. In particular, we show that node2vec random walk accelerates diffusion when walkers are designed to avoid both backtracking and visiting a neighbour of the previously visited node but do not avoid them completely.

Non-Fickian transport ubiquitously occurs across all scales within fractured geological media. Detailed characterization of non-Fickian transport through single fractures is thus critical for predicting the fate of solutes and other fluid-borne entities through fractured media. Our direct numerical simulations of solute transport through two-dimensional rough-walled fractures showed early arrival and heavy tailing in breakthrough curves (BTCs), which are salient characteristics of non-Fickian transport. Analyses for dispersion coefficients (DADE) using the standard advection-dispersion equation (ADE) led to errors which increased linearly with fracture heterogeneity. Estimated Taylor dispersion coefficients deviated from estimated DADE even at higher Peclet numbers. Alternatively, we used continuous time random walk (CTRW) model with truncated power law transition rate probability to characterize the non-Fickian transport. CTRW modeling markedly and consistently improved fits to the BTCs relative to those fitted with ADE solutions. The degree of deviation of transport from Fickian to non-Fickian is captured by the parameter β of the truncated power law. We found that β is proportional to fracture heterogeneity. We also found that the CTRW transport velocity can be predicted based on the flow velocity. Along with the ability to predict β, this is a major step toward prediction of transport through CTRW using measurable physical properties.

In Chapter 4, we obtained the governing equations of continuous-time random processes by taking a suitable joint limit Δ -→0 and τ -→0 of a lattice random walk, where Δ is the lattice spacing and τ the constant time between successive steps. The continuous-time processes so obtained are governed by stable densities (§4.3), with the Gaussian density representing the most typical case. In taking the continuum limit the lattice structure and the identity of individual steps are lost, and the analysis of many properties of the processes so obtained requires rather deep and subtle aspects of advanced probability theory, in contrast to the elegant conceptual simplicity of the original random walk. There is a less drastic way to obtain a continuous time process related to the ordinary random walk, without losing either the lattice structure or the notion of well-defined individual steps. This approach, the continuous time random walk model, owes its popularity in contemporary physics to Montroll and Weiss (1965). From a mathematical standpoint it is a particularly useful special case of the semi-Markov process (Levy 1954, Smith 1955, Pyke 1961, Feller 1964, Qinlar 1969, 1975).