Step Length Distribution Research Articles

Spatial organization in the step‐pool structure of a steep mountain stream (Vogelbach, Switzerland)

Spatial organization in the step‐pool structure of a 1.58 km long section of the Vogelbach, a steep mountain stream (average channel bed slope 0.17 m/m) in the Alpthal basin, Switzerland, is analyzed to evaluate the extent of randomness in the occurrence of steps and to isolate the potential hydraulic and topographic controls on step placement. Statistics of the observed step sequence are compared with those of random sequences generated by permutation from the observed step height distribution. Hydraulic and topographic controls on step placement are analyzed from step length and steepness distributions and relations between mean step properties and step height and channel gradient for both observed and random sequences. Results show that (1) observed step length distributions are statistically significantly different from randomly generated sequences, (2) step steepness is significantly different in observed data because of a positive correlation between mean step length and height and it remains fairly constant for all step sizes, (3) spatial organization in steps does not extend far beyond the nearest step, and (4) the influence of channel gradient on step properties is insignificant and highly variable, indicating that hydraulic rather than topographic controls are dominant for step placement in this stream. Although the Vogelbach is a steep stream on the boundary between a cascading and step‐pool morphology where we would expect randomness to dominate, spatial organization was nevertheless detected in many important aspects of the step‐pool geometry.

Complex animal distribution and abundance from memory-dependent kinetics

A realistic implementation of population kinetics, i.e., the transformation of individual movements to population mixing, intraspecific cohesion, site fidelity effects and other statistical mechanical aspects of population dynamics, is crucial for spatio-temporal modeling in ecology. Spatial memory effects on movements and homing are currently not realistically implemented in population dynamical modeling, which is based on an assumption of memory-free random walk and diffusion processes beyond fine-scaled behavior. We suggest a generic model where the animal relates to its familiar space strategically, by utilizing historic information for spatial memory mapping of its environment beyond its instant field of perception. Simulations illustrate how this mechanism may self-organize a scale-free, or fractal, habitat utilization even in a homogeneous environment, over a potentially large range of spatial scales. By tuning parameters for fractal dimension of step length distributions and return probability to previous locations, a continuum from a scale-free, memory-influenced displacement process to a classic, scale-specific and memory-free random walk-like process is achieved. Individual series of locations are superimposed with a varying degree of inter-connectedness to explore memory-influenced space use patterns in the context of population kinetics, i.e., by considering an ensemble of model individuals utilizing the same environment at the same time. A memory-dependent transition towards inter-individual dependency of space use is described as “intraspecific cohesion”.

Evidence on DNA slippage step-length distribution

A simple model based on a master equation is constructed in order to reveal the details of the mutational events modifying simple sequence repeats in the human genome, A database of simple repeats together with their flanking sequences comprising approximately 10(5) entries from all 24 human chromosomes was constructed. By aligning the pairs of fragments of sequences containing the repeat elements, the matrices that count the number of slippage events were evaluated. The counts were then used as a target to be reproduced by our theoretical model, in which the elongation and shortening of the repeats proceed through a mechanism in which the step lengths exhibit a decaying distribution in the form of an inverse power law rather than through one nucleotide extension or deletion, which was the most frequent supposition in previous studies.

Scale‐dependence of movement rates in stream invertebrates

We used analytical models and random walk simulations in a one‐dimensional habitat to study the scale‐dependence of migration rates in stream invertebrates. Our models predict that per capita migration rate is inversely proportional to patch length when patches are large compared to the scale of movements. When patches are small the scale‐dependence is weaker and primarily determined by the length of individual movements (steps) relative to patch size. Laboratory experiments using isopods (Asellus aquaticus L.) and mayfly nymphs (Baetis sp.) confirmed that the strength of the scale‐dependence decreased with increasing step length.For the case when step length distributions follow an exponential probability distribution, which is often the case for stream organisms, we provide a simple model that allows the scale‐dependence to be predicted from the mean step length. We fitted this model to published field data on drift densities at different downstream distances from a net that blocks the drift from upstream areas. Agreement between model and data was excellent in most cases. We then used already published data on the length of induced drift movements to predict the scale‐dependence that was observed in block experiments performed in the same system. Predicted and observed scale‐dependence showed very close agreement.We conclude that our models and published data on drift distances can be used to calculate the expected scale‐dependence of per capita emigration rates for a large number of taxa under a wide range of environmental conditions.

Lévy flight random searches in biological phenomena

There has been growing interest in the study of Lévy flights observed in the movements of biological organisms performing random walks while searching for other organisms. Here, we approach the problem of what is the best statistical strategy for optimizing the encounter rate between “searcher” and “target” organisms—either of the same or of different species—in terms of a limiting generalized searcher–target model (e.g., predator-prey, mating partner, pollinator–flower). In this context, we discuss known results showing that for fixed targets an inverse square density distribution of step lengths can optimize the encounter rate. For moving targets, we review how the encounter rate depends on whether organisms move in Lévy or Brownian random walks. We discuss recent findings indicating that Lévy walks confer a significant advantage for increasing encounter rates only when the searcher is larger or moves rapidly relative to the target, and when the target density is low.

The statistical mechanics of a concatenated polymer chain

The elastic and configurational properties of mesoscopic structures, which have been formed by linking a finite number of polymer loops to each other, are considered. For simplicity it is assumed that the linking numbers between pairs of loops provides a sufficient description of the topological structure. To conserve the topology the statistical mechanical calculation of the partition sum is restricted to the configuration space labelled by the specified linking numbers. An expansion of the partition sum is proposed where the linking numbers, corresponding to a particular structure, select a sub-set of terms from the expansion. The case of a linear concatenated chain of loops is studied and it is shown that the first non-zero contribution from this expansion is already able to describe the chain as a re-scaled Gaussian chain capable of supporting stress. This confirms that the approximation maintains the structural integrity of the chain. The effective step length and step length distribution function are also identified together with the elastic properties. The relationship of linking numbers to the Abelian Chern-Simons gauge field theory is used to illustrate essential parts of the calculation. The relevant term in the partition sum is represented in terms of Feynman diagrams describing the two-particle scattering amplitude for vector gauge bosons. It is shown that the expected re-scaled Gaussian chain result comes from the dominance in the s-channel of a massive scalar boson representing the structure of an individual loop in the chain.

Loops in one-dimensional random walks

Distribution of loops in a one-dimensional random walk (RW), or, equivalently, neutral segments in a sequence of positive and negative charges is important for understanding the low energy states of randomly charged polymers. We investigate numerically and analytically loops in several types of RWs, including RWs with continuous step-length distribution. We show that for long walks the probability density of the longest loop becomes independent of the details of the walks and definition of the loops. We investigate crossovers and convergence of probability densities to the limiting behavior, and obtain some of the analytical properties of the universal probability density.

Solvent diffusion in ordered macrofluids: A stochastic simulation study of the obstruction effect

An off-lattice random-flight simulation procedure is described which accurately predicts the obstruction factors for self-diffusion of small molecules in macrofluids. The simulation procedure, employing a continuous step length distribution that ensures rapid and accurate convergence, was validated by comparison with exact results for cylindrical and spherical obstructions on 2D and 3D lattices. The exact results were computed with Rayleigh’s multipole method, which also was used to derive a new analytical formula for the obstruction factor of parallel cylinders on a hexagonal lattice, of much higher accuracy than the commonly used approximations. Random-flight simulations were used to assess the accuracy of existing mean-field approximations for the obstruction factors of orientationally ordered nonspherical objects. Due to a near-cancellation of errors, the mean-field result accurately describes the obstruction effect on the trace of the diffusion tensor, as measured in isotropic systems, up to moderately high volume fractions. In contrast, the diffusion anisotropy, a sensitive indicator of microstructure in anisotropic fluids, is accurately predicted by mean-field theory only at low volume fractions.

Exact Results for Fully Directed Flights is s-Dimensions11The project supported by Natural Science Foundation of Jiangsu Educational Committee

The critical behaviour of s-dimensional fully directed flights (FDFs) is treated by a generating function method and an exact calculation method. Our results show that the exponents ν, ν∥ and ν⊥ of FDFs are independent of the spatial dimensionality and the concrete form of the probability distribution of step lengths. They are in agreement with the Laplacian transformation method and the Monte Carlo simulation study for fully directed Levy flights.

Step equalization in epitaxial growth: A noise limited process

We present results from the Monte Carlo simulation of terrace size evolution by vapor phase growth on vicinal surfaces. We show that as a function of the number of added monolayers, the standard deviation of the step length distribution rapidly falls to a constant equilibrium value. We argue that recent analytical and numerical work indicating that the standard deviation falls continuously is incorrect because of a statistical preaveraging of the evolution equations. This has experimental importance as it shows that increasing the number of monolayers beyond a certain limit has no effect on the step size distribution.

One-dimensional non-nearest-neighbor random walks in the presence of traps

A one-dimensional lattice random walk in the presence ofm equally spaced traps is considered. The step length distribution is a symmetric exponential. An explicit analytic expression is obtained for the probability that the random walk will be trapped at thejth trapping site.

Growth in a restricted solid on solid model with correlated noise

We introduce correlated growth into a restricted solid on solid model (RSOS), a stochastic deposition model with a constraint on neighboring height differences. A two-dimensional lattice model is used in which particles are deposited via horizontal Levy flight steps with a step length distribution exponentf. Though RSOS is in the same universality class as ballistic deposition for uncorrelated deposition, it appears to depart from it for strong correlations. Forf=1, the short-range limit is reached and both exponentsβ andχ, which describe the dependence of surface width on time and strip length, tend to 1. Forf>1 we retreat to an enhanced algorithm, searching for growth sites which become excessively rare. We find an unusual short-time dependence, butχ still tends to 1. The number of growth sitesG shows saturation forf<1, while forf⩾1 we observeG/L→0 as the strip lengthL increases. Finally, we test directly the relationship of noise-noise correlation strength tof, and find that a direct comparison between correlated growth models and theoretical predictions for growth with correlated noise is so far unjustified.

Fractal and nonfractal behavior in Levy flights

The d-dimentional space-continuous time-discrete Markovian random walk with a distribution of step lengths, which behaves like x−(α+d) with α&amp;gt;0 for large x, is studied. By studying the density–density correlation function of these walks, it is determined under what conditions the walks are fractal and when they are nonfractal. An ensemble average of walks is considered and the lower entropy dimension D of the set of stopovers of the walks in this ensemble is calculated, and D=min{2,α,d} is found. It is also found that the fractal nature of the walks is related to a finite value of the mean first passage time. The crossover of the correlation function from the fractal to nonfractal regimes is studied in detail. Finally, it is conjectured that these results for the lower entropy dimension apply to a wide class of symmetric Markov processes.

A Nonparametric, Distribution-Free Test for Serial Independence in Stock Returns

A fundamental statistical test of serial independence is developed and applied to daily stock returns. Let xt be the deviation of the daily return on a stock from its sample mean after any autocorrelation present has been removed. If xt is serially independent, then the cumulative sum of xt over time is the position of a one-dimensional random walk on a line. The empirical distribution of step lengths over a large sample allows the distribution of the largest absolute excursion in a T-step walk to be calculated by repeated simulation. The observed maximal excursions are found to be significantly smaller than one would expect, based on serial independence and the observed distribution of step lengths. It is concluded that these daily stock returns are not serially independent and that the market value of the corporations studied has a tendency to return to an interval around the trend value.

Chromatographic band broadening theory using a random walk with a step-length distribution.

Chromatographic band broadening theory using a random walk with a step-length distribution.

Cluster-particle aggregation with fractal (Levy flight) particle trajectories

The effects of particle trajectories on the structure of aggregates formed by cluster-particle aggregation have been explored by computer simulation. With the use of Levy-flight particle trajectories, clusters with a continuous range of effective dimensionalities between the limits of the Witten-Sander ($D\ensuremath{\simeq}\frac{5}{3}$ for $d=2$) and Vold-Sutherland models ($D\ensuremath{\simeq}2.0$ for $d=2$) can be generated. The effective dimensionality varies continuously with the exponent which describes the distribution of step lengths in the Levy-flight trajectory, but there does not seen to be a unique (1:1) relationship between the dimensionality of the trajectory and the dimensionality of the cluster.

Spitzer's condition for asymptotically symmetric random walk

If the step-length distribution function F for a random walk {Sn, n ≧ 0} is either continuous and symmetric or belongs to the domain of attraction of a symmetric stable law, then it is clear that the symmetric form of ‘Spitzer's condition' holds, i.e. The question considered in this note is whether or not (⋆) can hold for other random walks. The answer is in the affirmative, for we show that (⋆) holds for a large class of random walks for which F is neither symmetric nor belongs to any domain of attraction; all such random walks are asymptotically symmetric, in the sense that limx→∞ {F(–x)| 1 – F(x)} = 1, but we show by an example that this is not a sufficient condition for (⋆) to hold.

Spitzer's condition for asymptotically symmetric random walk

If the step-length distribution function F for a random walk {Sn, n ≧ 0} is either continuous and symmetric or belongs to the domain of attraction of a symmetric stable law, then it is clear that the symmetric form of ‘Spitzer's condition' holds, i.e. The question considered in this note is whether or not (⋆) can hold for other random walks. The answer is in the affirmative, for we show that (⋆) holds for a large class of random walks for which F is neither symmetric nor belongs to any domain of attraction; all such random walks are asymptotically symmetric, in the sense that lim x→∞ {F(–x)| 1 – F(x)} = 1, but we show by an example that this is not a sufficient condition for (⋆) to hold.

A stochastic model for particle sorting and related phenomena

A number of factors contribute to the variability in particle size distribution with position, or sorting, that is often observed in natural channels. Several of these factors are readily examined by using stochastic models of particle movement along a streambed. If particle position and particle size are treated as jointly distributed random variables, then sorting is reflected in the fact that the conditional distribution of size given position depends on the value of position. This conditional distribution is most easily derived by first postulating the form of the conditional distribution of position given size and of the marginal size distribution. Further, the effect of mixing particles from two or more distinct sources can be considered by examining the joint distribution of three random variables: size, position, and source. This is useful in predicting size distribution downstream from the point where a tributary enters a channel. Longitudinally varying transport conditions, which also contribute significantly to sorting, are conveniently modeled with a nonhomogeneous Poisson process. Here it is assumed that the particle motion consists of a sequence of steps of random length, each step followed by a rest period of random duration. It is the number of particle rest positions as a function of distance from the initial position that obeys a Poisson process, so the proposed model allows for systematic change of the step length and rest duration distributions with position downstream.

Stochastic Characteristics of Sediment Motions

The stochastic characteristics of the discrete movements of a sand grain under the action of flowing water are examined. Functional relationships between the step length (or rest duration) distributions and the probability of a sand grain making a specified number of steps (or rest durations) over a distance (or time) are derived and compared with the results from radioactive tracer experiments. A technique for determining the behavior of the density functions for step lengths and rest durations, involved the determination of the curvature of the autocorrelation function specially constructed from the step length and rest duration data.