Inversion problems for Fourier transforms of particle distributions

Jaeuk Kim, Ge Zhang, Frank H. Stillinger, Salvatore Torquato

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Collective coordinates in a many-particle system are complex Fourier components of the local particle density n(x) ≡ σN j=1 δ(x - rj), 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 (1D) 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 ⌈N/2⌉ smallest wavevectors is the minimal set of constraints for unique inversion, where ⌈·⌉ represents the ceiling function. This result provides useful groundwork to the inverse transform of collective coordinates in higherdimensional systems.

Original languageEnglish (US)
Article number113302
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number11
StatePublished - Nov 30 2018

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • random/ordered microstructures
  • structural correlations


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