Inversion Problems for Fourier Transforms of Particle Distributions

TL;DR

This paper investigates the minimal set of collective Fourier coordinates needed to uniquely invert and recover particle configurations in one-dimensional systems, revealing that roughly half of the smallest wavevectors suffice for unique inversion.

Contribution

It establishes that the smallest wavevector set of size approximately half the particles ensures unique inversion of collective coordinates to particle positions in 1D systems.

Findings

Minimal set of collective coordinates for unique inversion identified as about half the particles.

Numerical and theoretical analysis show the number of solutions depends on constraints and configurations.

Provides groundwork for inverse transforms in higher-dimensional particle systems.

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.