Spatial Form Factor for Point Patterns: Poisson Point Process, Coulomb Gas, and Vortex Statistics

TL;DR

This paper introduces a spatial form factor (SFF) to characterize point patterns in various point processes, analyzing its properties in different dimensions and applying it to vortex patterns in Bose-Einstein condensation.

Contribution

The paper defines the SFF for homogeneous Poisson processes, derives explicit formulas in multiple dimensions, and applies it to physical phenomena like vortex formation in ultracold atoms.

Findings

Explicit SFF expressions for Poisson processes in various dimensions

Connection between SFF and spacing distributions

Application of SFF to vortex pattern analysis in BEC

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.