Airy distribution: Experiment, large deviations, and additional statistics

Agranov, Tal; Zilber, Pini; Smith, Naftali R.; Admon, Tamir; Roichman, Yael; Meerson, Baruch

doi:10.1103/physrevresearch.2.013174

Cited by 33 publications

(47 citation statements)

References 61 publications

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“…(5), a power-law tail at large A and an essential singular behavior at A → 0 appear for all admissible n. In order to shed light on the nature of the essential singularity, we employ the optimal fluctuation method (OFM). The OFM has been successfully applied recently in several other problems, dealing with Brownian motion pushed to a large deviation regime by constraints [13][14][15][16][17]. Here we show that the OFM reproduces the essential singularity exactly by identifying the optimal, or most likely, path -a special trajectory of the Brownian motion that makes a dominant contribution to the PDF P n (A|x 0 ) at A → 0.…”

Section: Introductionmentioning

confidence: 70%

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Statistics of first-passage Brownian functionals

Majumdar¹,

Meerson²

2020

J. Stat. Mech.

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We study the distribution of first-passage functionals of the type A = t f 0 x n (t) dt where x(t) represents a Brownian motion (with or without drift) with diffusion constant D, starting at x0 > 0, and t f is the first-passage time to the origin. In the driftless case, we compute exactly, for all n > −2, the probability density Pn(A|x0) = Prob.(A = A). We show that Pn(A|x0) has an essential singular tail as A → 0 and a power-law tail ∼ A −(n+3)/(n+2) as A → ∞. The leading essential singular behavior for small A can be obtained using the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process in this limit. For the case with a drift toward the origin, where no exact solution is known for general n > −1, we show that the OFM successfully predicts the tails of the distribution. For A → 0 it predicts the same essential singular tail as in the driftless case. For A → ∞ it predicts a stretched exponential tail − ln Pn(A|x0) ∼ A 1/(n+1) for all n > 0. In the limit of large Péclet number Pe = µx0/(2D), where µ is the drift velocity toward the origin, the OFM predicts an exact large-deviation scaling behavior, valid for all A: − ln Pn(A|x0) Pe Φn z = A/Ā , whereĀ = x n+1

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Section: Introductionmentioning

confidence: 70%

“…B. Optimal fluctuation method explains essential singularity at A → 0 When applied to the Brownian motion, the OFM essentially becomes geometrical optics [13][14][15][16][17]. A natural starting point of the OFM is the probability of a Brownian path x (t), which is given, up to pre-exponential factors, by the Wiener's action, see e.g.…”

Section: A Exact Resultsmentioning

confidence: 99%

Statistics of first-passage Brownian functionals

Majumdar¹,

Meerson²

2020

J. Stat. Mech.

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“…In turn, one can define the correlation coefficient All the elements making up this expression have been derived earlier; the variance of the first passage time distribution can be computed from (10,11). The result is presented as a function of α in figure 3.…”

Section: The Area Statisticsmentioning

confidence: 99%

Statistics of the first passage area functional for an Ornstein–Uhlenbeck process

Kearney

Martin

2021

J. Phys. A: Math. Theor.

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We consider the area functional defined by the integral of an Ornstein–Uhlenbeck process which starts from a given value and ends at the time it first reaches zero (its equilibrium level). Exact results are presented for the mean, variance, skewness and kurtosis of the underlying area probability distribution, together with the covariance and correlation between the area and the first passage time. Among other things, the analysis demonstrates that the area distribution is asymptotically normal in the weak noise limit, which stands in contrast to the first passage time distribution. Various applications are indicated.

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“…The analysis of the far tail of P (x, t) was carried out in (Aghion et al, 2017), finding a spatial infinite (nonnormalized) density. It showed a relation between the laser cooling process and the problem of the distribution of random areas under Langevin excursions (Agranov et al, 2020;Barkai et al, 2014;Majumdar and Comtet, 2005). The latter is a constrained Langevin process starting and ending at p = 0, never crossing the origin within a given time interval (Fig.…”

Section: B From Sisyphus Friction To Lévy Walks In Position Spacementioning

confidence: 98%

Anomalous statistics of laser-cooled atoms in dissipative optical lattices

Afek¹,

Davidson²,

Kessler³

et al. 2021

Preprint

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In this Colloquium we discuss the anomalous kinetics of atoms in dissipative optical lattices, focusing on the "Sisyphus" laser cooling mechanism. The cooling scheme induces a friction force that decreases to zero for high atomic momentum, which in turn leads to unusual statistical features. We study, using a Fokker-Planck equation describing the semi-classical limit of the system, the shallow optical lattice regime where the momentum distribution of the particles is heavy-tailed and the spatial diffusion is anomalous. As the depth of the optical lattice is tuned, transitions in the dynamical properties of the system occur, for example a transition from Gaussian diffusion to a Lévy walk and the breakdown of the Green-Kubo formula for the diffusion constant. Rare events, in both the momentum and spatial distributions, are described by non-normalized states, with tools adapted from infinite ergodic theory. We present experimental observations and elementary explanations for the physical mechanisms of cooling that lead to these anomalous behaviors, comparing theory with available experimental and numerical data.

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Airy distribution: Experiment, large deviations, and additional statistics

Cited by 33 publications

References 61 publications

Statistics of first-passage Brownian functionals

Statistics of first-passage Brownian functionals

Statistics of the first passage area functional for an Ornstein–Uhlenbeck process

Anomalous statistics of laser-cooled atoms in dissipative optical lattices

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