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  • Open Access

Spatial form factor for point patterns: Poisson point process, Coulomb gas, and vortex statistics

Matteo Massaro1,* and Adolfo del Campo1,2

  • *Contact author: matteo.massaro@uni.lu
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Phys. Rev. Research 7, 023107 – Published 1 May, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.023107

Abstract

Point processes have broad applications in science and engineering. In physics, their use ranges from quantum chaos to statistical mechanics of many-particle systems. We introduce a spatial form factor (SFF) for the characterization of spatial patterns associated with point processes. Specifically, the SFF is defined in terms of the averaged even Fourier transform of the distance between any pair of points. We focus on homogeneous Poisson point processes and derive the explicit expression for the SFF in d-spatial dimensions. The SFF can then be found in terms of the even Fourier transform of the probability distribution for the distance between two independent and uniformly distributed random points on a d-dimensional ball, arising in the ball line picking problem. The relation between the SFF and the set of n-order spacing distributions is further established. The SFF is analyzed in detail for d=1,2,3 and in the infinite-dimensional case, as well as for the d-dimensional Coulomb gas, as an interacting point process. As a physical application, we describe the spontaneous vortex formation during Bose-Einstein condensation in finite time recently studied in ultracold atom experiments and use the SFF to reveal the stochastic geometry of the resulting vortex patterns. In closing, we also introduce a generalization of the SFF applicable to arbitrary sets in a metric space.

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Phys. Rev. Research 7, 023107– Published 1 May, 2025
  • Received 11 October 2024
  • Accepted 15 April 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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