Properties Of Walks Research Articles

Fractal space-times under the microscope: a renormalization group view on Monte Carlo data

The emergence of fractal features in the microscopic structure of space-time is a common theme in many approaches to quantum gravity. In this work we carry out a detailed renormalization group study of the spectral dimension d s and walk dimension d w associated with the effective space-times of asymptotically safe Quantum Einstein Gravity (QEG). We discover three scaling regimes where these generalized dimensions are approximately constant for an extended range of length scales: a classical regime where d s = d, d w = 2, a semi-classical regime where d s = 2d/(2 + d), d w = 2 + d, and the UV-fixed point regime where d s = d/2, d w = 4. On the length scales covered by three-dimensional Monte Carlo simulations, the resulting spectral dimension is shown to be in very good agreement with the data. This comparison also provides a natural explanation for the apparent puzzle between the short distance behavior of the spectral dimension reported from Causal Dynamical Triangulations (CDT), Euclidean Dynamical Triangulations (EDT), and Asymptotic Safety.

Dynamical Crossover in Complex Networks near the Percolation Transition

The return probability P 0 ( t ) of random walkers is investigated numerically for several scale-free fractal networks. Our results show that P 0 ( t ) is proportional to t - d s /2 with the non-integer spectral dimension d s as in the case of non-scale free fractal networks. We also study how the diffusion process is affected by the structural crossover from a fractal to a small-world architecture in a network near the percolation transition. It is elucidated that the corresponding dynamical crossover is scaled only by the unique characteristic time t ξ regardless of whether the network is scale free or not. In addition, the scaling relation d s = 2 D f / d w is found to be valid even for scale-free fractal networks, where D f and d w are the fractal and the walk dimensions. These results suggest that qualitative properties of P 0 ( t ) are irrelevant to the scale-free nature of networks.

Universality classes of first-passage-time distribution in confined media

We study the first-passage time (FPT) distribution to a target site for a random walker evolving in a bounded domain. We show that in the limit of large volume of the confining domain, this distribution falls into universality classes indexed by the walk dimension d(w) and the fractal dimension d(f) of the medium, which have been recently identified previously [Bénichou et al., Nat. Chem. 2, 472 (2010)]. We present in this paper a complete derivation of these universal distributions, discuss extensively the range of applicability of the results, and extend the method to continuous-time random walks. This analysis puts forward the importance of the geometry, and in particular the position of the starting point, in first-passage statistics. Analytical results are validated by numerical simulations, applied to various models of transport in disordered media, which illustrate the universality classes of the FPT distribution.

Distribution of first-passage times to specific targets on compactly explored fractal structures

The distribution of the first passage times (FPT) of a one-dimensional random walker to a target site follows a power law F(t)~t(-3/2). We generalize this result to another situation pertinent to compact exploration and consider the FPT of a random walker with specific source and target points on an infinite fractal structure with spectral dimension d(s)<2. We show that the probability density of the first return to the origin has the form F(t)~t(d(s)/2-2), and the FPT to a specific target at distance r follows the law F(r,t)~r(d(w)-d(f)) t(d(s)/2-2), where d(w) and d(f) are the walk dimension and the fractal dimension of the structure, respectively. The distance dependence of F(r,t) reproduces the one of the mean FPT of a random walk in a confined domain.

“I'm on Autopilot, I Just Follow the Route”: Exploring the Habits, Routines, and Decision-Making Practices of Everyday Urban Mobilities

Much urban and transport policy is based on a series of assumptions relating to predefined journeys informed by rational decision making. For example, the dominant focus of urban pedestrian policy is how the built environment is the primary influence on a person's decision to walk, with these decisions being made at a specific time and place somehow outside the practice of walking. Little attention is given to the importance of the processual and experiential dimensions of walking itself. This paper considers the ways in which journeys into, and through, the city unfold. Specific attention is drawn to how forms of habitual performance, such as practical knowledge and embodied habits and routines, assist in engaging with everyday urban movements. In the context of in-depth research conducted in London, the paper explores the habits and routines of urban pedestrian mobilities such as the daily commute, and in so doing, highlights the complexity and ongoing reconfiguration of these journeys beyond analysis informed by ‘rational’ intention. In contrast to much transport research that conceives of habit as an external force that somehow obstructs more positive sustainable travel behaviour, discussion focuses on the significance of how everyday urban walking unfolds. It is argued that this form of engagement makes it possible to understand habit as situated and part and parcel of the sequentially organised and occasioned performance of journeys on foot. This, in turn, enables the transformative potential of habitual behaviour to be realised. Throughout the paper consideration is given to specific narratives of everyday urban mobilities; the significance of how such practices are actually ‘talked about’; and how these accounts matter in engaging with the experiential dimensions of urban movement.

Walking in rhythms: place, regulation, style and the flow of experience

Following Henri Lefebvre, this article investigates the distinct rhythms of walking and the ways that it intersects with diverse temporalities and spaces. Because walking is practised and experienced in innumerable contexts, generalisations are problematic. Nevertheless, this article identifies some of the ways in which walking produces time-space and the experience of place. I subsequently discuss how walking is inevitably conditioned by multiple forms of regulation but possesses peculiar characteristics that always make these orderings of space and body contingent, facilitating immanent, often unexpected experiences. I further examine how walking is surrounded by notions of style that reproduce particular rhythms. Walking is inevitably, therefore, suffused with contending notions about how and where to walk, by ideals and conventions laid down by the powerful and not-so-powerful. However, I pay attention to the multiple phases and moments of walking. Both the conventions of walking and its unfolding, sensual and contingent apprehension are difficult to elucidate in academic prose, and luckily there are several challenging walking artists whose work highlights these issues and the rhythmic dimensions of walking. Accordingly, I draw upon the ‘textworks’ of Richard Long, Francis Alys's series of walks entitled Railings, and Jeremy Deller's 2009 Procession to elicit some of the rhythms of walking.

Fractal simulation of the resistivity and capacitance of arsenic selenide

The temperature dependences of the ac resistivity R and ac capacitance C of arsenic selenide were measured more than four decades ago [V. I. Kruglov and L. P. Strakhov, in Problems of Solid State Electronics, Vol. 2 (Leningrad Univ., Leningrad, 1968)]. According to these measurements, the frequency dependences are R ∝ ω−0.80±0.01 and ΔC ∝ ω−0.120±0.006 (ω is the circular frequency and ΔC is measured from the temperature-independent value C0). According to fractal-geometry methods, R ∝ ω1−3/h and ΔC ∝ ω−2+3/h, where h is the walk dimension of the electric current in arsenic selenide. Comparison of the experimental and theoretical results indicates that the walk dimensions calculated from the frequency dependences of resistivity and capacitance are hR = 1.67 ± 0.02 and hC = 1.60 ± 0.08, which are in agreement with each other within the measurement errors. The fractal dimension of the distribution of conducting sections is D = 1/h = 0.6. Since D < 1, the conducting sections are spatially separated and form a Cantor set.

Frequency and spatial characteristics of the electrophysical parameters of a living tree trunk

The method of vertical electrical sounding is applied to derive the spatial characteristic of the dc resistivity ρ of a living pine trunk, ρ = L-0.85, and the frequency characteristic of the ac resistance R of the same pine trunk, R = f-0.053. The results are simulated by the methods of fractal geometry, according to which ρ = L −h + 2 and \( R \sim f^{ - \tfrac{3} {h} + 1} \), where h is the walk dimension of the electric current in the trunk. From comparison between the experimental and theoretical results, it follows that h = 2.85 and fractal dimension D = 1/h = 0.35. Since D = 1, conducting layers form a Cantor set.

Heat kernel characterisation of Besov-Lipschitz spaces on metric measure spaces

We give a heat-kernel characterisation of the Besov-Lipschitz spaces Lip (α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of Grigor’yan et al. (Trans Am Math Soc 355:2065–2095, 2008), Hu and Zahle (Studia Math 170:259–281, 2005), Pietruska-Paluba (Stoch Stoch Rep 67:267–285, 1999; 70:153–164, 2000), which were valid for \({\alpha = \frac{d_w}{2}, p = 2, q = \infty}\), where dw is the walk dimension of the space X.

One-dimensional drug release from finite Menger sponges: In silico simulation

The purpose of this work was to evaluate the consequences of the spatial distribution of components in pharmaceutical matrices type Menger sponge on the drug release kinetic from this kind of platforms by means of Monte Carlo computer simulation. First, six kinds of Menger sponges (porous fractal structures) with the same fractal dimension, d f = 2.727 , but with different random walk dimension, d w ∈ [ 2.149 , 3.183 ] , were constructed as models of drug release device. Later, Monte Carlo simulation was used to describe drug release from these structures as a diffusion-controlled process. The obtained results show that drug release from Menger sponges is characterized by an anomalous behavior: there are important effects of the microstructure anisotropy, and porous structures with the same fractal dimension but with different topology produce different release profiles. Moreover, the drug release kinetic from heteromorphic structures depends on the axis used to transport the material to the external medium. Finally, it was shown that the number of releasing sites on the matrix surface has a significant impact on drug release behavior and it can be described quantitatively by the Weibull function.

Random walk on topologically biased anisotropic percolation clusters and transport properties

Unbiased random walks are performed on topologically biased anisotropic percolation clusters (APC). Topologically biased APCs are generated using suitable anisotropic percolation models. New walk dimensions are found to characterize the anisotropic behaviour of the unbiased random walk on the biased topology. Critical properties of electro and magneto conductivities are characterized estimating respective dynamical critical exponents. A dynamical scaling theory relating dynamical and static critical exponents has been developed. The dynamical critical exponents satisfy the scaling relations within error bar.

On the interaction of geometry, analysis and stochastics on a body

AbstractIt is reasonable to expect that the geometrical feature of a body has influence to spectral asymptotics of its “natural” Laplacian as well as to the behavior of its “natural” Brownian motion. In fact, such an interaction can be expressed by a so–called “Einstein relation” implicating Hausdorff, spectral and walk dimension. These quantities express geometric, analytic and stochastic aspects of a set. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A multifractal decomposition according to rate of returns

AbstractFor one‐dimensional simple symmetric random walk, the Hausdorff and packing dimensions of sets of sample paths with prescribed rate of returns to the origin are determined. This gives a multifractal decomposition of the underlying sample space. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Investigation of the timed‘Up &amp; Go’test in children

The timed‘Up & Go’test (TUG) is a test of basic or functional mobility in adults which has rarely been used in children. Functional mobility was defined for this study as an individual's ability to manoeuvre his or her body capably and independently to accomplish everyday tasks. Reliability and validity of TUG scores were examined in 176 children without physical disabilities (94 males, 82 females; mean age 5y 9mo [SD 1y 8mo]; range 3 to 9y) and in 41 young people with physical disabilities due to cerebral palsy or spina bifida (20 males, 21 females; mean age 8y 11mo [SD 4y 3mo], range 3 to 19y). Mean TUG score for children without physical disability was 5.9s (SD 1.3). Reliability of the TUG test was high, with intraclass correlation coefficients (ICC) of 0.89 within session, and 0.83 for test-retest reliability. Mean score of the group aged 3 to 5 years was significantly higher (6.7s SD 1.2) than that of the older group (5.1s, SD 0.8;p=0.001). Scores in the younger group reduced significantly over a 5-month follow-up period (p=0.001), indicating that the TUG was responsive to change. Within-session reliability of the TUG in young people with disabilities was very high (ICC=0.99). There were significant differences in TUG scores between children classified at levels I, II, and III of the Gross Motor Function Classification System (p=0.001). TUG scores showed a moderate negative correlation with scores on the Standing and Walking dimensions of the Gross Motor Function Measure (n=22, rho=-0.52,p=0.012). There was no significant difference in TUG scores between typically developing male and female children. The TUG can be used reliably in children as young as 3 years using the protocol described in this paper. It is a meaningful, quick, and practical objective measure of functional mobility. With further investigation, the TUG is potentially useful as a screening test, an outcome measure in intervention studies for young people with disabilities, a measure of disability, and as a measure of change in functional mobility over time.

Anomalous diffusion in two‐dimensional Euclidean and prefractal geometrical models of heterogeneous porous media

Disordered systems are known to induce anomalous diffusion. This phenomenon may be important for environmental applications such as contaminant transport and nutrient availability. However, few studies have investigated anomalous diffusion in this context. In particular, the relationship between pore space geometry and anomalous diffusion is not well understood. We report on numerical simulations of solute diffusion within the water‐filled pore spaces of two‐dimensional geometrical models of heterogeneous porous media. Euclidean and mass, pore, and pore‐solid prefractal lattices were used to generate random pore networks with varying porosity (ϕ) and lacunarity (L). The objectives were to investigate the effects of ϕ and L on the solute random walk dimension (dw) and to identify which of these models best represents a natural porous medium. Solute diffusion was simulated using a stochastic cellular automaton based on the “myopic ant” algorithm. Estimates of dw ≫ 2 occurred with increasing frequency as ϕ → 0, indicating scale dependency in the standard diffusion coefficient at low porosities. The relationship between dw and ϕ for the mass and pore‐solid prefractal lattices was the closest to that for natural 2‐D systems (i.e., soil thin sections). The presence of large, interconnected pore spaces (L → 1) at low porosities reduced the intensity of anomalous diffusion (dw → 2). A power law relationship based on the product of dw and L explained &gt;96% of the total variation in ϕ regardless of the type of lattice considered. The potential predictive capability of this approach for natural porous media deserves further investigation.

Simple transient random walks in one-dimensional random environment: the central limit theorem

We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the central limit theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.

Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal

This report introduces a novel method to characterize the diffusion-time dependence of the diffusion-weighted magnetic resonance (MR) signal in biological tissues. The approach utilizes the theory of diffusion in disordered media where two parameters, the random walk dimension and the spectral dimension, describe the evolution of the average propagators obtained from q-space MR experiments. These parameters were estimated, using several schemes, on diffusion MR spectroscopy data obtained from human red blood cell ghosts and nervous tissue autopsy samples. The experiments demonstrated that water diffusion in human tissue is anomalous, where the mean-square displacements vary slower than linearly with diffusion time. These observations are consistent with a fractal microstructure for human tissues. Differences observed between healthy human nervous tissue and glioblastoma samples suggest that the proposed methodology may provide a novel, clinically useful form of diffusion MR contrast.

Monte carlo simulation of diffusion-limited drug release from finite fractal matrices

How fast can drug molecules escape from a controlled matrix-type release system? This important question is of both scientific and practical importance, as increasing emphasis is placed on design considerations that can be addressed only if the physical chemistry of drug release is better understood. In this work, this problem is studied via Monte Carlo computer simulations. The drug release is simulated as a diffusion-controlled process. Six types of Menger sponges (all having the same fractal dimension, df = 2.727, but with different values of random walk dimension, dw ∈ [2.028, 2.998]) are employed as models of drug delivery devices with the aim of studying the consequences of matrix structural properties (characterized by df and dw) on drug release performance. The results obtained show that, in all cases, drug release from Menger sponges follows an anomalous behavior. Finally, the influence of the matrix structural properties on the drug release profile is quantified.

Asymptotic Behaviour of the Simple Random Walk on the 2-dimensional Comb

We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical ones, respectively. The limit process is obtained. With similar techniques the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb.

Evaluation of the longer-term use of the David Hart Walker Orthosis by children with cerebral palsy: a 3-year prospective evaluation

Purpose. To evaluate a walking device, the David Hart Walker Orthosis (HW), that was designed to allow children with severe cerebral palsy to ambulate with hands-free support.Method. A pre-/post-test prospective one-group study evaluated outcomes three years after receiving the HW. Physical therapy assessment, parent interview and satisfaction questionnaire provided details on outcomes.Results. The HW remained the sole walking device for 13 of 20 children at 3 years. Six of seven children who discontinued use were over 12 years old and had outgrown its maximum size. Twelve of 13 children who still used the HW were assessed. GMFM Stand and Walk Dimension mean score increases of about 3% points for the 1 – 3-year follow-up were not significant (P > 0.16). Timed walk scores were unchanged. Steering ability gains were demonstrated on a directional mobility assessment (from 12.0 to 27.9% [P = 0.02]). Despite its eventual height limitations, parents considered HW use to be worthwhile (mean satisfaction = 8.2/10).Conclusions. Sixty-five percent of the children continued to use the HW as their sole walking device, and demonstrated improved ability overall to manoeuvre it during functional ambulation. The primary reason for discontinuation was inability to accommodate taller children.