TY - JOUR
T1 - Inversion problems for Fourier transforms of particle distributions
AU - Kim, Jaeuk
AU - Zhang, Ge
AU - Stillinger, Frank H.
AU - Torquato, Salvatore
N1 - Funding Information:
This work was supported partially by the National Science Foundation under Grant No. CBET-1701843.
Publisher Copyright:
© 2018 IOP Publishing Ltd and SISSA Medialab srl.
PY - 2018/11/30
Y1 - 2018/11/30
N2 - 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.
AB - 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.
KW - random/ordered microstructures
KW - structural correlations
UR - http://www.scopus.com/inward/record.url?scp=85057616663&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85057616663&partnerID=8YFLogxK
U2 - 10.1088/1742-5468/aae84c
DO - 10.1088/1742-5468/aae84c
M3 - Article
AN - SCOPUS:85057616663
SN - 1742-5468
VL - 2018
JO - Journal of Statistical Mechanics: Theory and Experiment
JF - Journal of Statistical Mechanics: Theory and Experiment
IS - 11
M1 - 113302
ER -