Probability distribution of distance in a uniform ellipsoid: Theory and applications to physics

Parry, Michelle; Fischbach, Ephraim

doi:10.1063/1.533249

Cited by 23 publications

(23 citation statements)

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“…The projection of R ij onto the Z-axis is uniformly with the auxiliary constant a = 27 √ 3/(8π). The probability density function for the random variable Y follows from the density [111,112] f R (r) = …”

Section: Discussionmentioning

confidence: 99%

Spectral backbone of excitation transport in ultracold Rydberg gases

2014

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The spectral structure underlying excitonic energy transfer in ultra-cold Rydberg gases is studied numerically, in the framework of random matrix theory, and via self-consistent diagrammatic techniques. Rydberg gases are made up of randomly distributed, highly polarizable atoms that interact via strong dipolar forces. Dynamics in such a system is fundamentally different from cases in which the interactions are of short range, and is ultimately determined by the spectral and eigenvector structure. In the energy levels' spacing statistics, we find evidence for a critical energy that separates delocalized eigenstates from states that are localized at pairs or clusters of atoms separated by less than the typical nearest-neighbor distance. We argue that the dipole blockade effect in Rydberg gases can be leveraged to manipulate this transition across a wide range: As the blockade radius increases, the relative weight of localized states is reduced. At the same time, the spectral statistics-in particular, the density of states and the nearest neighbor level spacing statistics-exhibits a transition from approximately a 1-stable Lévy to a Gaussian orthogonal ensemble. Deviations from random matrix statistics are shown to stem from correlations between inter-atomic interaction strengths that lead to an asymmetry of the spectral density and profoundly affect localization properties. We discuss approximations to the self-consistent Matsubara-Toyozawa locator expansion that incorporate these effects.

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Section: Discussionmentioning

confidence: 99%

Spectral backbone of excitation transport in ultracold Rydberg gases

2014

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“…We shall derive, for any 1 d  and any 1 nd and for circumspheres of () n d C , the joint probability density function (pdf) of the length  and of the circumradius  as well as the associated marginal pdf's.For reasons discussed below, the case of two random points   the length 12 AA has been calculated during the last hundred years by a variety of methods which yield different formal expressions [6, 15-16, 18-20, 24-25, 30, 36, 40, 44, 46, 48-49, 52, 58, 60-61, 63].The pdf's of the distance between two points randomly distributed in the inside of spheres or of ellipsoids find numerous applications. For instance, they were shown to simplify the calculation of self-energies of spherically symmetric matter distributions interacting by means of radially symmetric two-body potentials [48][49]. These calculations were extended to ellipsoids as a first step towards convex bodies whose shapes deviate from spherical.…”

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confidence: 99%

Circumspheres of sets of n + 1 random points in the d-dimensional Euclidean unit ball (1 ≤ n ≤ d)

Caër

2017

Journal of Mathematical Physics

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In the d -dimensional Euclidean space, any set of 1 n  independent random points, uniformly distributed in the interior of a unit ball of center O , determines almost surely a circumsphere of center C and of radius         and 1 T    . The rv  has a simple geometrical meaning which appears in figure 1a. A non-degenerate triangle ' O CM   0, 0   , where M is any point of the circumsphere except B , yields '' O M O C CM     . If M coincides with B , then the triangle becomes a segment and ' OB   . Therefore, a circumsphere is contained in the unit ball if and only if r   . We notice, en passant, that the rv's  , C  ,  are also relevant for circumspheres which cut   1 1 d S . We shall derive, for any 1 d  and any 1 nd and for circumspheres of () n d C , the joint probability density function (pdf) of the length  and of the circumradius  as well as the associated marginal pdf's.For reasons discussed below, the case of two random points   the length 12 AA has been calculated during the last hundred years by a variety of methods which yield different formal expressions [6, 15-16, 18-20, 24-25, 30, 36, 40, 44, 46, 48-49, 52, 58, 60-61, 63].The pdf's of the distance between two points randomly distributed in the inside of spheres or of ellipsoids find numerous applications. For instance, they were shown to simplify the calculation of self-energies of spherically symmetric matter distributions interacting by means of radially symmetric two-body potentials [48][49]. These calculations were extended to ellipsoids as a first step towards convex bodies whose shapes deviate from spherical. García-Pelayo [24] derived the distance pdf in ellipsoids as an integral he applied to a study of the shape of the earth. Other physical applications of distance pdf's include in particular the use of double electron-electron resonance to study spherical aggregates with shell structure [34] and the field of wireless networks whose properties are strongly influenced by distances between nodes [45][46]58]. Finally, it is worth mentioning the connection between distance pdf's and pdf's of random chord length in convex bodies, which depends on the considered secant randomness (see for instance [13]). The chord length pdf's apply for instance in the fields of neutronics and of reactor physics [19, 26, 35, 37, 41-42, 52, 57, 63]. Extensions of the previous problem include for instance the determination of the mean distance between a reference point and its n th neighbour among a collection of N points uniformly distributed in a hypersphere or in a hypercube of unit volumes in a d -dimensional Euclidean space [8]. Applying the results of the latter authors, Kowalski [38] gave a geometrical interpretation to a generalization of a distribution used to represent pion distribution in hadronic production models. Circumcircles and circumspheres play an important role in computational geometry. Domains of all kinds are meshed with Delaunay triangulations and Voronoi tessellations are constructed from them. Direct ap...

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“…As discussed in Refs. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], these results are of interest as tools in mathematical physics, and have numerous applications in other fields as well. Specifically, it was demonstrated in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…In two recent papers [1,2], geometric probability techniques were developed to calculate the functions P 3 (s) which describe the probability density of finding a random distance s separating two random points distributed in a uniform sphere and in a uniform ellipsoid. As discussed in Refs.…”

Section: Introductionmentioning

confidence: 99%

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Random distance distribution for spherical objects: general theory and applications to physics

Fischbach

2002

J. Phys. A: Math. Gen.

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A formalism is presented for analytically obtaining the probability density function, Pn(s), for the random distance s between two random points in an n-dimensional spherical object of radius R. Our formalism allows Pn(s) to be calculated for a spherical n-ball having an arbitrary volume density, and reproduces the well-known results for the case of uniform density. The results find applications in stochastic geometry, computational science, molecular biological systems, statistical physics, astrophysics, condensed matter physics, nuclear physics, and elementary particle physics. As one application of these results, we propose a new statistical method obtained from our formalism to study random number generators in n-dimensions used in Monte Carlo simulations.

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Probability distribution of distance in a uniform ellipsoid: Theory and applications to physics

Cited by 23 publications

References 13 publications

Spectral backbone of excitation transport in ultracold Rydberg gases

Spectral backbone of excitation transport in ultracold Rydberg gases

Circumspheres of sets of n + 1 random points in the d-dimensional Euclidean unit ball (1 ≤ n ≤ d)

Random distance distribution for spherical objects: general theory and applications to physics

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