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

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.

17 pages, 7 figures

References in corpus (28)

• Spontaneous vortices in the formation of Bose-Einstein condensates
• Black Holes and Random Matrices
• Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques
• Hyperuniform States of Matter
• Universality of Phase Transition Dynamics: Topological Defects from Symmetry Breaking
• Critical Dynamics of Spontaneous Symmetry Breaking in a Homogeneous Bose gas
• Structural defects in ion crystals by quenching the external potential: the inhomogeneous Kibble-Zurek mechanism
• The stochastic Gross-Pitaevskii equation II
• Scrambling the spectral form factor: unitarity constraints and exact results
• Causality in Condensates: Grey Solitons as Remnants of BEC Formation
• Defect formation beyond Kibble-Zurek mechanism and holography
• Causality and non-equilibrium second-order phase transitions in inhomogeneous systems
• Kibble-Zurek universality in a strongly interacting Fermi superfluid
• Full Counting Statistics of Topological Defects After Crossing a Phase Transition
• Inhomogeneous Kibble-Zurek mechanism: vortex nucleation during Bose-Einstein condensation
• Universal breakdown of Kibble-Zurek scaling in fast quenches across a phase transition
• Universal Statistics of Vortices in a Newborn Holographic Superconductor: Beyond the Kibble-Zurek Mechanism
• Numerical method for evolving the Projected Gross-Pitaevskii equation
• Defect Saturation in a Rapidly Quenched Bose Gas
• Low-Dimensional Stochastic Projected Gross-Pitaevskii Equation
• Topological defect formation in a phase transition with tunable order
• Exactly-solvable system of one-dimensional trapped bosons with short and long-range interactions
• Locality of Spontaneous Symmetry Breaking and Universal Spacing Distribution of Topological Defects Formed Across a Phase Transition
• Universal Early Coarsening of Quenched Bose Gases
• Exact ground states of quantum many-body systems under confinement
• Universal Vortex Statistics and Stochastic Geometry of Bose-Einstein Condensation
• Suppression of Spontaneous Defect Formation in Inhomogeneous Bose Gases
• Decomposing the Spectral Form Factor

Cited by in corpus (1)

• Crystalline Spectral Form Factors