The Pearson walk with shrinking steps in two dimensions

Serino, C. A.; Redner, S.

doi:10.1088/1742-5468/2010/01/p01006

Cited by 12 publications

(22 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…While cosmology is an obvious example, there has been recent interest in nonconstant metric also in thin sheets [15][16][17]. Our results are applicable to the formation of stochastic patterns and structures in a very general setting, including diffusion processes with time-dependent diffusion rate (i.e., temperature) [18][19][20][21], in cosmologically expanding space [22], or on a biologically growing substrate.In particular, we consider self-affine space-time trajectories of particles under spatially homogeneous but time dependent metric, and map those into more easily tractable systems with constant metric. The mapping depends only on the local scale invariance exponent of the trajectories, and works directly for local interactions which do not involve a length scale, such as annihilation or coagulation of point particles.…”

mentioning

confidence: 88%

mentioning

confidence: 88%

“…For example one can study exponentially increasing domains, which is analogous to structures with exponentially decreasing diffusivity. These have been studied in detail for single random walks [18][19][20][21] and are used in simulated annealing [33]. In n + 1 dimensions, where n is the spatial dimensionality, our method applies directly if the scale invariance holds in all spatial directions i = 1, .…”

mentioning

confidence: 99%

See 2 more Smart Citations

Scale-invariant growth processes in expanding space

et al. 2013

View full text Add to dashboard Cite

Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting particles, and their large scale behavior depends on the overall growth geometry. We establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries, which preserves the local scale invariance and is independent of other properties such as the dimensionality. This relation generalizes standard conformal transformations as the natural symmetry of self-affine growth processes. We illustrate our main result numerically for various structures of coalescing Lévy flights and fractional Brownian motions, including also branching and finite particle sizes. One of the main benefits of this approach is a full, explicit description of the asymptotic statistics in expanding domains, which are often nontrivial and random due to amplification of initial fluctuations.

show abstract

mentioning

confidence: 88%

mentioning

confidence: 88%

mentioning

confidence: 99%

See 1 more Smart Citation

Scale-invariant growth processes in expanding space

et al. 2013

View full text Add to dashboard Cite

show abstract

“…appearing in equations (50a) and (50b) can be thought of as a 'random walk' with a linearly 'shrinking' time-dependent step length (see, e.g., [94,95] for other examples of such random walks). In this case, the product of cosines in equations (52a) and (52b) can be explicitly written down as…”

mentioning

confidence: 99%

Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

et al. 2018

View full text Add to dashboard Cite

The power spectral density (PSD) of any time-dependent stochastic process X t is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X t over an infinitely large observation time T, that is, it is defined as an ensemble-averaged property taken in the limit  ¥ T . A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation time T. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is a fluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.

show abstract

“…Now we know random walk is purely diffusive. But still for verification we have calculated mean square displacement S 2 for the dimensions (1,2,3,4,5,6,7,8,9,10) which is proportional to time (t) that reveals diffusive behavior (fig(1)). The distribution of absolute displacement after time steps (N t = 1000) in d-dimensional continuum is plotted in (fig4) where it is seen that the distributions are non-monotonic and unimodal and as we increase the dimension, the most probable value i.e maximum probability of finding the walker at a distance (S m ) from the starting point is shifting towards higher value.…”

Section: Model and Resultsmentioning

confidence: 99%

Statistics of Projected Motion in One Dimension of a D-Dimensional Random Walker

Chattopadhyay¹,

Acharyya²

2018

View full text Add to dashboard Cite

We are studying the motion of a random walker in generalised d-dimensional continuum with unit step length (up to 10 dimensions) and its projected one dimensional motion numerically. The motion of a random walker in lattice or continuum is well studied in statistical physics but what will be the statistics of projected one dimensional motion of higher dimensional random walker is yet to be explored. Here in this paper, addressing this particular type of problem, we have showed that the projected motion is diffusive irrespective of any dimension, however, the diffusion rate is changing inversely with dimensions. As a consequence, it can be predicted that for the one dimensional projected motion of infinite dimensional random walk, the diffusion rate will be zero. This is an interesting result, at least pedagogically, which implies that though in infinite dimensions there is a diffusion but its one dimensional projection is motionless. At the end of the discussion we are able to make a good comparison between projected one dimensional motion of generalised d-dimensional random walk with unit step length and pure one dimensional random walk with random step length varying uniformly between −h to h where h is a 'step length renormalizing factor'.

show abstract

The Pearson walk with shrinking steps in two dimensions

Cited by 12 publications

References 34 publications

Scale-invariant growth processes in expanding space

Scale-invariant growth processes in expanding space

Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

Statistics of Projected Motion in One Dimension of a D-Dimensional Random Walker

Contact Info

Product

Resources

About