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

Tu, Shu-Ju; Fischbach, Ephraim

doi:10.1088/0305-4470/35/31/303

Cited by 42 publications

(32 citation statements)

References 36 publications

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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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“…The proposed model calculates distance cdf as a function of three independent variables, the networks radii and the separation distance. Therefore, the cdf may be obtained from the intersection volume of two solids; see, for example, [18][19][20].…”

Section: The Nodal Distance Distribution Modelmentioning

confidence: 99%

“…The third semi-axis of the ellipsoid is related to the others and it is set empirically (further discussion follows below). Finally, we locate the centers of the sphere and the ellipsoid at distance D [19] (recall that in [1], the centers of the two circles were placed at distance equal to the distance between the single point and the network's center).…”

Section: The Nodal Distance Distribution Modelmentioning

confidence: 99%

A Geometric Method for Computing the Nodal Distance Distribution in Mobile Networks

Baltzis

2011

PIER

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Abstract-This paper presents a geometrically based method for the calculation of the node-to-node distance distribution function in circular-shaped networks. In our approach, this function is obtained from the intersection volume of a sphere and an ellipsoid. The method is valid for both overlapping and non-overlapping networks. Simulation results and comparisons with methods in the literature demonstrate the efficacy of the approach. The relation between networks geometric parameters and distance statistics is explored. As an application example, we model distance-dependent path loss and investigate the impact of channel characteristics and networks size on signal absorption. The aforementioned model is a useful and lowcomplexity tool for system-level modeling and simulation of mobile communication systems.

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“…With regard to the distribution of the distance between points chosen uniformly at random to lie within a Euclidean d-dimensional sphere, the work in [6] derives the probability density function for the distance between two points. In order keep the exposition simple, we will confine ourselves to the 2-dimensional Euclidean sphere, i.e.…”

Section: Random Points In D-dimensional Spheres and Emergent Propertiesmentioning

confidence: 99%

“…For randomly chosen points within a circle of radius R, the following is a direct consequence of the work in [6]:…”

Section: Random Points In D-dimensional Spheres and Emergent Propertiesmentioning

confidence: 99%

The digital territory: a mathematical model of the concept and its properties

Kameas

Stamatiou²

2006

2nd IET International Conference on Intelligent Environments (IE 06)

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In this paper we attempt to define a mathematical model for the formal study of the concept of Digital Territories (DT) and its properties. We describe a simple DT model and employ the first order language of graphs and random graph properties in order to define and study properties of DTs that emerge as soon as the entities (within the DT) population reaches certain threshold points. Our aim is to show that the DT concept can be formalized and studied using mathematical tools that are within reach of any researcher in the DT domain in order to describe and study properties of interest.

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

Cited by 42 publications

References 36 publications

Spectral backbone of excitation transport in ultracold Rydberg gases

Spectral backbone of excitation transport in ultracold Rydberg gases

A Geometric Method for Computing the Nodal Distance Distribution in Mobile Networks

The digital territory: a mathematical model of the concept and its properties

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