Inversion Problems for Fourier Transforms of Particle Distributions

Abstract

Collective coordinates in a many-particle system are complex Fourier components of the particle density, and often provide useful physical insights. However, given collective coordinates, it is desirable to infer the particle coordinates via inverse transformations. In principle, a sufficiently large set of collective coordinates are equivalent to particle coordinates, but the nonlinear relation between collective and particle coordinates makes the inversion procedure highly nontrivial. Given a "target" configuration in one-dimensional Euclidean space, we investigate the minimal set of its collective coordinates that can be uniquely inverted into particle coordinates. For this purpose, we treat a finite number $M$ of the real and/or the imaginary parts of collective coordinates of the target configuration as constraints, and then reconstruct "solution" configurations whose collective coordinates satisfy these constraints. Both theoretical and numerical investigations reveal that the number of numerically distinct solutions depends sensitively on the chosen collective-coordinate constraints and target configurations. From detailed analysis, we conclude that collective coordinates at the $\lceil \frac{N}{2}\rceil$ smallest wavevectors is the minimal set of constraints for unique inversion, where $\lceil \cdot \rceil$ represents the ceiling function. This result provides useful groundwork to the inverse transform of collective coordinates in higher-dimensional systems.

arXiv comment21 pages (including references), 10 figures

References in corpus (16)

• Local Density Fluctuations, Hyperuniformity, and Order Metrics
• Hyperuniform States of Matter
• Designer disordered materials with large complete photonic band gaps
• Classical Disordered Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous Materials
• Transport Phase Diagram and Anderson Localization in Hyperuniform Disordered Photonic Materials
• Ensemble Theory for Stealthy Hyperuniform Disordered Ground States
• Designing Disordered Hyperuniform Two-Phase Materials with Novel Physical Properties
• Transport, Geometrical and Topological Properties of Stealthy Disordered Hyperuniform Two-Phase Systems
• Hyperuniform disordered phononic structures
• Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations
• Collective Coordinate Control of Density Distributions
• Point processes and stochastic displacement fields
• The Perfect Glass Paradigm: Disordered Hyperuniform Glasses Down to Absolute Zero
• Ground states of stealthy hyperuniform potentials. II. Stacked-slider phases
• Realizability of point processes
• Classical many-particle systems with unique disordered ground states

Cited by in corpus (1)

• Light in correlated disordered media