Winding statistics of a Brownian particle on a ring

Kundu, Anupam; Comtet, Alain; Majumdar, Satya N.

doi:10.1088/1751-8113/47/38/385001

Cited by 8 publications

(22 citation statements)

References 41 publications

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“…Here we will confine ourselves to evaluating the joint distribution P(θ, T ) and the achieved winding angle distribution p(θ) in the limits of very low and very high mortality, where we can rely on the previously obtained long-and short-time asymptotics of P 1 (θ, T ). The long-time asymptotic of P 1 (θ, T ) corresponds to the strong inequalities (DT ) 1/2 R, L. In this limit P 1 (θ, T ) does not depend on L [14,[17][18][19][20]:…”

Section: Story 3: Winding Angle Distributionmentioning

confidence: 92%

Mortal Brownian motion: Three short stories

Meerson

2019

Int. J. Mod. Phys. B

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Mortality introduces an intrinsic time scale into the scale-invariant Brownian motion. This fact has important consequences for different statistics of Brownian motion. Here we are telling three short stories, where spontaneous death, such as radioactive decay, puts a natural limit to "lifetime achievements" of a Brownian particle. In story 1 we determine the probability distribution of a mortal Brownian particle (MBP) reaching a specified point in space at the time of its death. In story 2 we determine the probability distribution of the area A = T 0 x(t)dt of a MBP on the line. Story 3 addresses the distribution of the winding angle of a MBP wandering around a reflecting disk in the plane. In stories 1 and 2 the probability distributions exhibit integrable singularities at zero values of the position and the area, respectively. In story 3 a singularity at zero winding angle appears only in the limit of very high mortality. A different integrable singularity appears at a nonzero winding angle. It is inherited from the recently uncovered singularity of the short-time large-deviation function of the winding angle for immortal Brownian motion. * Electronic address: meerson@mail.huji.ac.il arXiv:1903.03963v2 [cond-mat.stat-mech]

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Section: Story 3: Winding Angle Distributionmentioning

confidence: 92%

Mortal Brownian motion: Three short stories

Meerson

2019

Int. J. Mod. Phys. B

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“…Very recently, winding of Ornstein-Uhlenbeck processes [29] and of stable processes [5] were studied, including analysis of large scale asymptotics and limit laws. Some other relatively recent studies of winding with physical applications include [8,13,14,20].…”

Section: Background and Motivationmentioning

confidence: 99%

The winding of stationary Gaussian processes

Buckley

Feldheim

2017

Probab. Theory Relat. Fields

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This paper studies the winding of a continuously differentiable Gaussian stationary process f : R → C in the interval [0, T ]. We give formulae for the mean and the variance of this random variable. The variance is shown to always grow at least linearly with T , and conditions for it to be asymptotically linear or quadratic are given. Moreover, we show that if the covariance function together with its second derivative are in L 2 (R), then the winding obeys a central limit theorem. These results correspond to similar results for zeroes of real-valued stationary Gaussian functions by Malevich, Cuzick, Slud and others.

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“…More generally, studying Brownian paths with topological constrains (here a hole) is an important subject in many fields of physics (physics of polymers [50,51], fluxlines in superconductors [52]) and mathematics [53,54,55]. Besides, taboo processes in a circular annulus extended in some way the recent results exposed in the article of Kundu, Comtet and Majumdar concerning the properties of a Brownian particle on a ring [56].…”

Section: Two Dimensional Taboo Processmentioning

confidence: 99%

Sweetest taboo processes

Mazzolo¹

2018

J. Stat. Mech.

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Brownian dynamics play a key role in understanding the diffusive transport of micro particles in a bounded environment. In geometries containing confining walls, physical laws determine the behavior of the random trajectories at the boundaries. For impenetrable walls, imposing reflecting boundary conditions to the Brownian particles leads to dynamics described by reflecting stochastic differential equations. In practice, these stochastic differential equations as well as their refinements are quite challenging to handle, and more importantly, many physical processes are better modeled by processes conditioned to stay in a prescribed bounded region. In the mathematical literature, these processes are known as taboo processes, and despite their simplicity, at least compared to the reflecting stochastic differential equations approach, are surprisingly not much exploited in physics. This paper explores some aspect of taboo processes and other constrained processes in simple geometries: Interval in one dimension, circular annulus in two dimensions, hollow sphere in three dimensions, and more. In particular, for the two-dimensional taboo process in a circular annulus, the Gaussian behavior of the stochastic angle is established.

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Winding statistics of a Brownian particle on a ring

Cited by 8 publications

References 41 publications

Mortal Brownian motion: Three short stories

Mortal Brownian motion: Three short stories

The winding of stationary Gaussian processes

Sweetest taboo processes

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