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

TL;DR

This work introduces the Spatial Form Factor (SFF) as a 1D diagnostic for spatial point patterns, linking it to the even Fourier transform of the pairwise distance distribution. It provides exact SFF expressions for binomial/Poisson point processes on a d-dimensional ball, expressed via hypergeometric functions, and relates SFF to n-th nearest-neighbor spacing distributions, drawing connections to quantum-chaos spectral form factors. The paper extends the framework to interacting systems via a Coulomb gas, deriving a high-temperature linear-in-β correction in 1D and presenting higher-dimensional results through Monte Carlo simulations, and demonstrates the SFF’s utility in characterizing vortex formation during Bose-Einstein condensation. Overall, the SFF offers a versatile geometric tool for analyzing stochastic point patterns and their spacing statistics, with broad implications for quantum chaos, defect formation, and beyond, including potential generalizations to arbitrary metric spaces.

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.

Figures (7)

• Figure 1: Structure factor of a point system on a regular lattice and BPP. Panels (a) and (b) show a system of $N = 100$ points on a square lattice and its corresponding structure factor $S(\mathbf{k})$, determined numerically and plotted along $\mathbf{k} = (k, 0)$. Panels (c) and (d) depict a specific realization of a BPP with $N = 100$ points within a unit radius disk, and the structure factor of the corresponding process, plotted along $\mathbf{k} = (k, 0)$. In panel (d), the thin solid line represents the analytical expression given by Eq. (\ref{['S(K)_BPP']}), while the thicker transparent line is obtained by numerically computing $S(\mathbf{k})$ using a Monte Carlo simulation, averaging over 100 realizations of the process.
• Figure 2: Comparison of the structure factor $S(k)$ along $\mathbf{k} = k \hat{x}$ and the spatial form factor (SFF) for various point patterns. Panels (a), (b), and (c) illustrate a regular square lattice, its corresponding $S(k)$, and SFF, respectively. Panels (d), (e), and (f) show a Penrose tiling (P3) with its $S(k)$ and SFF. Panels (g), (h), and (i) present a single realization of a Poisson point process (PPP), along with its $S(k)$ and SFF. All patterns are generated with identical numbers of points $N$ and the same density. The structure factor signals crystalline or regular order through the presence of spikes. By contrast, the SFF provides complementary information regarding the spacing distribution and is suited for the characterization of disordered systems.
• Figure 3: Spatial form factor of a BPP on a $d$-dimensional ball with unit radius. Panels (a), (b), and (c) correspond respectively to $d=1,2,3$. In each of the three cases, the solid thin line represents the analytic expression given in Eq. (\ref{['finalsimplifiedSFFdBall']}), while the transparent thicker line corresponds to the SFF computed numerically in a Monte Carlo simulation with $N=40$ points, averaging over 300 randomly generated configurations.
• Figure 4: Different components of the $\mathrm{SFF}$ for a BPP of $N=1500$ points. Panels (a), (b), (c) and (d) correspond respectively to the first, second, fifth and tenth component of the $\mathrm{SFF}$ for the 1D process on a segment of length $2R=100$. Panels (e), (f), (g) and (h) show the same components of the $\mathrm{SFF}$ for the 2D process on a disk of radius $R=50$. In all cases, the thin solid line corresponds to the analytical expression provided in Table (\ref{['tab:gff_components']}), while the thick transparent line corresponds to the $\mathrm{SFF}^{(n)}$ computed numerically by a Monte Carlo simulation, averaging over $300$ trials. For convenience, in all the panels, we have plotted the rescaled $\mathrm{SFF}^{(n)}$ with the total number of points.
• Figure 5: Comparison of the $\rm SFF$ for a BPP, corresponding to $\beta=0$, and a Coulomb gas with $\beta=0.04$ in 1D. The system consists of $N = 20$ particles confined to the segment $[-1, 1]$, with each particle carrying a charge $q=+1$. The red curve represents the analytic expression for the Coulomb gas, given by Eq. (\ref{['final_result_GFF_Coulomb_beta_linear']}), while the black curve corresponds to the analytic $\rm{SFF}$ of a BPP, as derived in Sec. \ref{['GFFPPP']}. A logarithmic scale is used on the $y$-axis to highlight the differences between the two curves.