Dimensional Random Walk Research Articles

Anomalous diffusion in simulations of pumping tests on fractal lattices

Abstract The recent interest in fractals in the geosciences literature has led to several proposed theoretical models for the hydraulic testing of fractured-rock systems that exhibit a fractal-like geometric structure. There is, however, no agreement on the correct form of the resulting model equations. In order to gain some insight into the range of possible behaviours to be expected from pumping tests on such systems, as well as the type of theoretical models needed, extensive simulations of pressure diffusion for transient groundwater flow, modelled by random walks on both deterministic and random fractal lattices were performed. For simplicity, the focus was on measurements of the random-walk dimension for generalized Sierpinski carpets, a proposed model for porous and fractured media. In addition to the expected anomalous slow down in diffusion in fractals as measured by the random-walk dimension, the simulations show further novel and unexpected anomalous behaviour due to the presence of internal boundaries at all scales. None of the proposed theoretical models for pumping tests on fractals appears consistent with all of the observed anomalous behaviours. The simulations suggest that interpretation of experimental pumping tests in terms of well-defined non-integer dimensions can be difficult, even when finite-size effects are negligible.

Poisson trees, succession lines and coalescing random walks

We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S⊂ R d . The algorithm is independent of the origin of coordinates. We show that (1) the graph has one topological end – that is, from any point there is exactly one infinite self-avoiding path; (2) the graph has a unique connected component if d=2 and d=3 (a tree) and it has infinitely many components if d⩾4 (a forest); (3) in d=2 and d=3 we construct a bijection between the points of the Poisson process and Z using the preorder-traversal algorithm. To construct the graph we interpret each point in S as a space-time point (x,r)∈ R d−1× R . Then a ( d−1)-dimensional random walk in continuous time continuous space starts at site x at time r. The first jump of the walk is to point x′, at time r′> r, ( x′, r′)∈ S, where r′ is the minimal time after r such that | x− x′|<1. All the walks jumping to x′ at time r′ coalesce with the one starting at ( x′, r′). Calling ( x′, r′)= α( x, r), the graph has vertex set S and edges {( s, α( s)), s∈ S}. This enables us to shift the origin of S ∘= S∪{0} (the Palm version of S) to another point in such a way that the distribution of S ∘ does not change (to any point if d=2 and d=3; point-stationarity).

Non Uniform Random Walks

Given $\epsilon _i ∈ [0,1)$ for each $1 &lt; i &lt; n$, a particle performs the following random walk on $\{1,2,...,n\:\}$par If the particle is at $n$, it chooses a point uniformly at random (u.a.r.) from $\{1,...,n-1\}$. If the current position of the particle is $m (1 &lt; m &lt; n)$, with probability $\epsilon _m$ it decides to go back, in which case it chooses a point u.a.r. from $\{m+1,...,n\}$. With probability $1-\epsilon _m$ it decides to go forward, in which case it chooses a point u.a.r. from $\{1,...,m-1\}$. The particle moves to the selected point. What is the expected time taken by the particle to reach 1 if it starts the walk at $n$? Apart from being a natural variant of the classical one dimensional random walk, variants and special cases of this problemarise in Theoretical Computer Science [Linial, Fagin, Karp, Vishnoi]. In this paper we study this problem and observe interesting properties of this walk. First we show that the expected number of times the particle visits $i$ (before getting absorbed at 1) is the same when the walk is started at $j$, for all $j &gt; i$. Then we show that for the following parameterized family of $\epsilon 's: \epsilon _i = \frac{n-i}{n-i+ α · (i-1)}$,$1 &lt; i &lt; n$ where $α$ does not depend on $i$, the expected number of times the particle visits $i$ is the same when the walk is started at $j$, for all $j &lt; i$. Using these observations we obtain the expected absorption time for this family of $\epsilon 's$. As $α$ varies from infinity to 1, this time goes from $Θ (log n) to Θ (n)$. Finally we studythe behavior of the expected convergence timeas a function of $\epsilon$ . It remains an open question to determine whether this quantity increases when all $\epsilon 's$ are increased. We give some preliminary results to this effect.

A complexity theory model in science education problem solving: random walks for working memory and mental capacity.

The present study examines the role of limited human channel capacity from a science education perspective. A model of science problem solving has been previously validated by applying concepts and tools of complexity theory (the working memory, random walk method). The method correlated the subjects' rank-order achievement scores in organic-synthesis chemistry problems with the subjects' working memory capacity. In this work, we apply the same nonlinear approach to a different data set, taken from chemical-equilibrium problem solving. In contrast to the organic-synthesis problems, these problems are algorithmic, require numerical calculations, and have a complex logical structure. As a result, these problems cause deviations from the model, and affect the pattern observed with the nonlinear method. In addition to Baddeley's working memory capacity, the Pascual-Leone's mental (M-) capacity is examined by the same random-walk method. As the complexity of the problem increases, the fractal dimension of the working memory random walk demonstrates a sudden drop, while the fractal dimension of the M-capacity random walk decreases in a linear fashion. A review of the basic features of the two capacities and their relation is included. The method and findings have consequences for problem solving not only in chemistry and science education, but also in other disciplines.

Anisotropic diffusion and correlation analysis.

A method of statistical analysis of the correlation between two given scale invariant sequences is proposed. The relation between the fractal dimension of a two-dimensional random walk, generated with jumps derived from the signals, and the scaling exponents of the sequences is investigated, and a well-defined relation is found in the case of statistically independent signals. The method of analysis, whose performance is described for the case of an intermittent map, might represent a new tool for the study of the correlation between coupled complex systems.

Fractal dimension of the active zone for a p-doped poly(3,4-ethylenedioxythiophene) modified electrode towards a ferrocene probe

The electrochemical properties of p-doped poly(3,4-ethylenedioxythiophene) (PEDOT) modified electrodes obtained by anodic electrooxidation of 3,4-ethylenedioxythiophene on a platinum electrode have been studied. The heterogeneous electron transfer concerning the oxidation of the ferrocene occurs at the p-doped PEDOT film ∣ solution interface. Because of the interface heterogeneity, an anomalous diffusion is observed and the kinetics of the electron transfer during the oxidation of the ferrocene have been examined in terms of the fractal concept. A reaction dimension or an active zone that is defined as the operative fractal dimension is determined by means of rotating disk electrode measurements. A functional equation corresponding to the limiting current density at different rotating rates is given. A fractal dimension of random walk d w of 1.80 is obtained. A representative plot that allows us to normalize the measured current density with respect to the anomalous diffusion is proposed. In acetonitrile, the oxidized heavy p-doped PEDOT film may be viewed as an amorphous organic metal which induces an anomalous diffusion process in its vicinity for diffusion—convection limited reactions.

Image numérique et compacité d'opérateurs de composition sur un espace de Hilbert de séries de Dirichlet

In this Note, we study the numerical range and compacity of some composition operators, defined on a Hilbert space of Dirichlet series introduced by Hedenmalm, Linqvist and Seip. We show that most often, zero is the interior of this numerical range. The study of compacity exhibits a phenomenon which recalls Polya's theorem on random walks in dimension d: if the length of the symbol of the operator is d+1, and if its image is non-trivial, then the operator is non-compact if d=1; compact, non Hilbert–Schmidt, if d=2; Hilbert–Schmidt if d⩾3. To cite this article: C. Finet et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 325–328.

Renormalizations of Branching Random Walks in Equilibrium

We study the $d$-dimensional branching random walk for $d \gt 2$. This process has extremal equilibria for every intensity. We are interested in the large space scale and large space-time scale behavior of the equilibrium state. We show that the fluctuations of space and space-time averages with a non-classical scaling are Gaussian in the limit. For this purpose we use the historical process, which allows a family decomposition. To control the distribution of the families we use the concept of canonical measures and Palm distributions.

Symmetric random walk in random environment in one dimension

We give a new proof of the central limit theorem for one dimensional symmetric random walk in random environment. The proof is quite elementary and natural. We show the convergence of the generators and from this we conclude the convergence of the process. We also investigate the hydrodynamic limit (HDL) of one dimensional symmetric simple exclusion in random environment and prove stochastic convergence of the scaled density field. The macroscopic behaviour of this field is given by a linear heat equation. The diffusion coefficient is the same as that of the corresponding random walk.

The pore structure of Sierpinski carpets

In this paper, a new method is developed to investigate the pore structure of finitely and even infinitely ramified Sierpinski carpets. The holes in every iteration stage of the carpet are described by a hole-counting polynomial. This polynomial can be computed iteratively for all carpet stages and contains information about the distribution of holes with different areas and perimeters, from which dimensions governing the scaling of these quantities can be determined. Whereas the hole area is known to be two dimensional, the dimension of the hole perimeter may be related to the random walk dimension.

Renormalization of the Voter Model in Equilibrium

We consider the $d$-dimensional voter model for $d \geq 3$. Our interest is the large scale limit of the equilibrium state of the voter model, where we prove the $d = 3$ results of [1] for $d \geq 4$, which turn out to be of a different nature than for $d = 3$. For this purpose we use the historical process.We establish some surprising facts about the Green’s function of random walks in dimension $d \geq 4$,which lead to the different features in $d = 3$ versus $d \geq 4$. Secondly, we prove an analogous result for the voter model on the hierarchical group.

Modelling porous structures by repeated Sierpinski carpets

Porous materials such as sedimentary rocks often show a fractal character at certain length scales. Deterministic fractal generators, iterated upto several stages and then repeated periodically, provide a realistic model for such systems. On the fractal, diffusion is anomalous, and obeys the law 〈 r 2〉∼ t 2/ d w , where 〈 r 2〉 is the mean square distance covered in time t and d w >2 is the random walk dimension. The question is: How is the macroscopic diffusivity related to the characteristics of the small scale fractal structure, which is hidden in the large-scale homogeneous material? In particular, do structures with same d w necessarily lead to the same diffusion coefficient at the same iteration stage? The present paper tries to shed some light on these questions.

Effect of a forbidden site on ad-dimensional lattice random walk

We study the effect of a single excluded site on the diffusion of a particle undergoing random walk in a d-dimensional lattice. The determination of the characteristic function allows to find explicitly the asymptotical behaviour of physical quantities such as the particle average position (drift) and the mean square deviation. Contrarily to the one-dimensional case, where the average coordinate diverges at infinite times as t**1/2 and where the diffusion constant D is changed due to the impurity, the effects of the latter are shown to be much less important in higher dimensions: for d=842, the position is simply shifted by a constant and the diffusion constant remains unaltered although dynamical corrections (logarithmic for d=2) still occur. Finally, the continuum space version of the model is analyzed; it is shown that d=1 is the lower dimensionality above which all the effects of the forbidden site are irrelevant.

Critical properties of a branched polymer growth model.

We study the branched polymer growth model (BPGM) introduced by Lucena et al. [Phys. Rev. Lett. 72, 230 (1994)] in two dimensions. First the BPGM was simulated in very large lattices with concentrations of impurities q=0 and q=0.2. The scaling of the mass in chemical space gives accurate estimates of the critical branching probabilities b(c) and of the chemical dimensions Dc at criticality, improving previous results. Estimates of the fractal dimension D(F) at criticality are consistent with a universal value along the critical line. Our results for q=0 suggest small deviations of Dc and D(F) from the percolation values. We also simulated the BPGM in finite lattices of lengths between L=32 and L=512 for the same concentrations q. Using finite-size scaling techniques, we confirm the previous estimates of D(F) and the universality along the critical line, and obtain the correlation exponent nu=1.43+/-0.06. It proves that the BPGM is not in the same universality class of percolation in two dimensions. Finally, we simulate random walks on the critical polymers grown in very large lattices with q=0 and q=0.2, and obtain the random walk dimension Dw and the spectral dimension Ds. Dw is larger and Ds is smaller than the corresponding values in critical percolation clusters, due to the lower connectivity of the polymers. The scaling relation Ds=2D(F)/Dw is not satisfied, as observed in other tree-like structures.

Resistance scaling and random walk dimensions for finitely ramified Sierpinski carpets

We present a new algorithm to calculate the random walk dimension of finitely ramified Sierpinski carpets. The fractal structure is interpreted as a resistor network for which the resistance scaling exponent is calculated using Mathematica. A fractal form of the Einstein relation, which connects diffusion with conductivity, is used to give a numerical value for the random walk dimension.

AN EFFICIENT IMPLEMENTATION OF THE EXACT ENUMERATION METHOD FOR RANDOM WALKS ON SIERPINSKI CARPETS

In the following, we present a highly efficient algorithm to iterate the master equation for random walks on effectively infinite Sierpinski carpets, i.e. without surface effects. The resulting probability distribution can, for instance, be used to get an estimate for the random walk dimension, which is determined by the scaling exponent of the mean square displacement versus time. The advantage of this algorithm is a dynamic data structure for storing the fractal. It covers only a little bit more than the points of the fractal with positive probability and is enlarged when needed. Thus the size of the considered part of the Sierpinski carpet need not be fixed at the beginning of the algorithm. It is restricted only by the amount of available computer RAM. Furthermore, all the information which is needed in every step to update the probability distribution is stored in tables. The lookup of this information is much faster compared to a repeated calculation. Hence, every time step is speeded up and the total computation time for a given number of time steps is decreased.

Ergodicity and Stability of Stochastic Processes

Harris Markov chains ergodicity of non-Harris chains stochastic recursive sequences Markov chains in random environment multidimensional Markov processes two dimensional random walks three (and more) dimensional random walks continuous time Markov processes ergodicity of communication and queueing networks.

Spectral features in emissions from random amplifying media

We have examined the phenomena of line-narrowing and bichromatic emission in random amplifying media by means of Monte Carlo simulations. We explain the various reported experimental features through a model that includes only the stimulated emission and self-absorption of photons performing a three dimensional random walk inside the amplifying medium. We present the results for all three regimes of scattering, L<l ∗ , L∼l ∗ and L>l ∗ .

An investigation into the determination of critical dimension of self-avoiding random walk on a d-dimensional simplex fractal

By setting up the relevant recursion relations and by doing exact and approximate calculations, we show that there is no critical dimension in a self-avoiding random walk on a simplex fractal.

Transmission and scattering of a Lorentz gas on a slab

We perform numerical scattering experiments on a Lorentz array of disks centered on a triangular lattice with L columns and study its transmission and reflection properties. In the finite horizon case, the motion of the particles may be modeled as simple one dimensional random walks with absorbing walls for which the scaling of the transmission and reflection coefficients are known, and agree with those found numerically. In the infinite horizon case the analogy with a simple diffusive process is no longer valid. In this case we compare our results with those expected for a one dimensional Levy walk, again with absorbing walls, for which logarithmic corrections to the scaling relations appear. The scaling with L and the symmetry properties of the forward and the backward are also studied, and some of their salient features are discussed.