Random Changes In Direction Research Articles

Cell motility in cancer, crucial events, criticality, and Lévy walks

The analysis of glioblastoma (GB) cell locomotion and its modeling inspired by Lévy random walks is presented herein. We study such walks occurring on a two-dimensional plane where the walk is similar to the motion of a bird flying with a constant velocity, but with random changes of direction in time. The intelligence of the bird is signaled by the instantaneous changes of flying direction, which become invisible in the time series obtained by projecting the 2D walk either on the x-axis or y-axis. We establish that the projected 1D time series share the statistical complexity of time series frequently used to monitor physiological processes, shedding light on the role of crucial events (CE-s) in pathophysiology. Such CE-s are signified by abrupt changes of flying direction which are invisible in the 1D physiological time series. We establish a connection between the complex scaling index δ generated by the CE-s through μR=2−δ, where μR is the inverse power law index of the probability density function of the time interval between consecutive failures of the process of interest. We argue that the identification of empirical indices along with their theoretical relations afford important measures to control cancer.

Analysis of Supersonic Jets in Flapping Screech Mode based on Time-Resolved Three-Dimensional Velocity Measurement

The time-resolved three-dimensional velocity measurement of an underexpanded jet is performed using time-resolved microphone and non-time-resolved particle image velocimetry (PIV) measurements. Flapping screech jets at the nozzle pressure ratio (NPR) of 2.97 was measured and analyzed using the proposed approach. It was found that the dominant azimuthal mode at the present case is flapping mode. The screech tones are strongly associated with the first two proper orthogonal decomposition modes of the dominant azimuthal modes of the microphone data. The azimuthal characteristics of these modes are helical mode, and the rotation direction is opposite. These findings are qualitatively consistent with the previous results. The flapping structure rotates over a wide range of temporal scales, with random changes in direction of rotation.

Telegraph random evolutions on a circle

We consider the random evolution described by the motion of a particle moving on a circle alternating the angular velocities ±c and changing rotation at Poisson random times, resulting in a telegraph process over the circle. We study the analytic properties of the semigroup it generates as well as its probability distribution. The asymptotic behavior of the wrapped process is also discussed in terms of circular Brownian motion. Besides, it is possible to derive a stochastic model for harmonic oscillators with random changes in direction and we give a diffusive approximation of this process. Furthermore, we introduce some extensions of the circular telegraph model in the asymmetric case and for non-Markovian waiting times as well. In this last case, we also provide some asymptotic results.

Movement Path Tortuosity in Free Ambulation: Relationships to Age and Brain Disease.

Ambulation is defined by duration, distance traversed, number and size of directional changes, and the interval separating successive movement episodes; more complex measures of ambulation can be created by aggregating these features. This review article of published findings defines random changes in direction during movement as "movement path tortuosity" and relates tortuosity to the understanding of cognitive impairments of persons of all ages. Path tortuosity is quantified by subjecting tracking data to fractal analysis, specifically Fractal Dimension (Fractal D), which ranges from a value of 1 when the movement path is perfectly straight to a value of 2 when the movement path is random, resembling the "drunkard's walk." The review elucidates the mathematical assumptions underlying Fractal D, its use in the analysis of movements of free ranging animals, and its application to the study of cognitive impairment and the prediction of falls in older adults. We conclude Fractal D offers a reliable, valid, sensitive, and easily interpreted real-time longitudinal measure of unrestricted movement path tortuosity unaffected by mobility aid use.

Goldstein-Kac telegraph processes with random speeds: Path probabilities, likelihoods, and reported Lévy flights.

The Goldstein-Kac telegraph process describes the one-dimensional motion of particles with constant speed undergoing random changes in direction. Despite its resemblance to numerous real-world phenomena, the singular nature of the resultant spatial distribution of each particle precludes the possibility of any a posteriori empirical validation of this random-walk model from data. Here we show that by simply allowing for random speeds, the ballistic terms are regularized and that the diffusion component can be well-approximated via the unscented transform. The result is a computationally efficient yet robust evaluation of the full particle path probabilities and, hence, the parameter likelihoods of this generalized telegraph process. We demonstrate how a population diffusing under such a model can lead to non-Gaussian asymptotic spatial distributions, thereby mimicking the behavior of an ensemble of Lévy walkers.

Fault current Characterization Based on Fuzzy Algorithm for DOCR Application

Penetration of distribution generation (DG) into power system might disturb the existing fault diagnosis system. The detection of fault, fault classification, and random changes of direction of fault current cannot always be monitored and determined via on-line by conventional fault diagnosis system due to DG penetration. In this paper, a fault current characterization which based on fuzzy logic algorithm (FLA) is proposed. Fault detection, fault classification, and fault current direction are extracted after processing the measurement result of three-phase line current. The ability of fault current characterization based on FLA is reflected in directional overcurrent relay (DOCR) model. The proposed DOCR model has been validated in microgrid test system simulation in Matlab environment. The simulation result showed accurate result for different fault location and type. The proposed DOCR model can operate as common protection device (PD) unit as well as unit to improve the effectiveness of existing fault diagnosis system when DG is present.

Stirring by squirmers

We analyse a simple ‘Stokesian squirmer’ model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress (Phys. Lett. A, vol. 374, 2010, p. 3487), where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that, for the viscous case, the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate non-zero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly swimming squirmers exhibit probability distribution functions with exponential tails and a short-time superdiffusive regime, as found previously by several authors. In our case, the exponential tails are due to ‘sticking’ near the stagnation points on the squirmer's surface.

Scaling the fractional advective–dispersive equation for numerical evaluation of microbial dynamics in confined geometries with sticky boundaries

Microbial motility is often characterized by ‘run and tumble’ behavior which consists of bacteria making sequences of runs followed by tumbles (random changes in direction). As a superset of Brownian motion, Levy motion seems to describe such a motility pattern. The Eulerian (Fokker–Planck) equation describing these motions is similar to the classical advection–diffusion equation except that the order of highest derivative is fractional, α ∈ (0, 2]. The Lagrangian equation, driven by a Levy measure with drift, is stochastic and employed to numerically explore the dynamics of microbes in a flow cell with sticky boundaries. The Eulerian equation is used to non-dimensionalize parameters. The amount of sorbed time on the boundaries is modeled as a random variable that can vary over a wide range of values. Salient features of first passage time are studied with respect to scaled parameters.

An extension of generalized Taylor dispersion in unbounded homogeneous shear flows to run-and-tumble chemotactic bacteria

In the absence of flow, the biased random walk of bacteria such as Escherichia coli is modeled by straight runs punctuated by random changes in direction, tumbles. We model chemotaxis by allowing the tumble rate of a run to depend on the component of swimming velocity in the direction of the chemoattractant gradient. In the well-studied situation of weak bias in tumble rate, bacteria disperse over a diffusive time scale, and the evolution of density satisfies the classic Keller–Segel advection-diffusion equation. In this paper, we model swimming bacteria being advected and rotated by an unbounded homogeneous shear flow. The flow field alters the trajectories of individuals and thus affects the macroscopic dispersion of a population. We adapt the formal framework of generalized Taylor dispersion theory to make it applicable for run-and-tumble bacteria with an arbitrary bias in tumble rate. This enables us to obtain a macroscopic description of the dispersion of bacteria. For the particular case of simple shear flow, we calculate explicitly the effect of flow on the diffusion tensor and mean swimming velocity.

Grassmann-valued processes for the Weyl and the Dirac equations.

The time evolution of a massless particle satisfying the Weyl equation is described as a stochastic process on a space of Grassmann variables, in close formal analogy with the use of Brownian motion for Schr\"odinger evolution. The Grassmann process is then combined with a Poisson process previously used for the evolution of Dirac electrons. (In that process electrons propagate as massless left- or right-handed particles with random changes in direction occurring at an average rate given by the particle mass.) Electron motion is thus given as an expectation over the two processes and spatial position supersymmetrically acquires contributions from sums of products of Grassmann variables.

The meaning of the term ‘klinokinesis’

The original definition of the term klinokinesis was based on an account by Ullyott (1936) of the behaviour of the flatworm, Dendrocoelum lacteum, an account recently shown to be unsound. Recently, the behaviour of Escherichia coli and of Salmonella typhimurium in chemical gradients has been found to conform to the definition, so the term can continue in use. But no uncomplicated case has been found in metazoans, though the manoeuvres of klinokinesis may make large contributions to the behaviour of some of them, even though it may be complicated and obscured by other types of reaction, such as orthokinesis. What is needed is mathematical treatment sufficiently thorough to enable the existence of the reaction in metazoans to be ascertained. There may well be other systems of manoeuvres using random changes of direction and these should be sought and investigated.

An invariance principle for a class of 𝑑-dimensional polygonal random functions

A class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space ( d ≥ 1 ) (d \geq 1) undergoing random changes of direction at random times while maintaining constant speed. The changes of direction are determined by random orthogonal matrices that are irreducible in the sense of not having an almost surely invariant nontrivial subspace if d ≥ 2 d \geq 2 , and not being almost surely nonnegative if d = 1 d = 1 . An invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved. The limit process has mean zero and its covariance matrix function is given explicitly. It is shown that when the random changes of direction satisfy an appropriate condition the limit process is Brownian motion. This invariance principle includes central limit theorems for the plane, with special distributions of the random times and direction changes, that have been proved by M. Kac, V. N. Tutubalin and T. Watanabe by methods different from ours. The proof makes use of standard methods of the theory of weak convergence of probability measures, and special results due to P. Billingsley and B. Rosén, the main problem being how to apply them. For this, renewal theoretic techniques are developed, and limit theorems for sums of products of independent identically distributed irreducible random orthogonal matrices are obtained.