Abstract
We present local distributed, stochastic algorithms for alignment in selforganizing particle systems (SOPS) on twodimensional lattices, where particles occupy unique sites on the lattice, and particles can make spatial moves to neighboring sites if they are unoccupied. Such models are abstractions of programmable matter, composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational capabilities. We consider oriented particle systems, where particles are assigned a vector pointing in one of q directions, and each particle can compute the angle between its direction and the direction of any neighboring particle, although without knowledge of global orientation with respect to a fixed underlying coordinate system. Particles move stochastically, with each particle able to either modify its direction or make a local spatial move along a lattice edge during a move. We consider two settings: (a) where particle configurations must remain simply connected at all times and (b) where spatial moves are unconstrained and configurations can disconnect.
Our algorithms are inspired by the Potts model and its planar oriented variant known as the planar Potts model or clock model from statistical physics. We prove that for any q ≥ 2, by adjusting a single parameter, these selforganizing particle systems can be made to collectively align along a single dominant direction (analogous to a solid or ordered state) or remain nonaligned, in which case the fraction of particles oriented along any direction is nearly equal (analogous to a gaseous or disordered state). In the connected SOPS setting, we allow for two distinct parameters, one controlling the ferromagnetic attraction between neighboring particles (regardless of orientation) and the other controlling the preference of neighboring particles to align. We show that with appropriate settings of the input parameters, we can achieve compression and expansion, controlling how tightly gathered the particles are, as well as alignment or nonalignment, producing a single dominant orientation or not. While alignment is known for the Potts and clock models at sufficiently low temperatures, our proof in the SOPS setting are significantly more challenging because the particles make spatial moves, not all sites are occupied, and the total number of particles is fixed.
BibTeX  Entry
@InProceedings{kedia_et_al:LIPIcs.APPROX/RANDOM.2022.14,
author = {Kedia, Hridesh and Oh, Shunhao and Randall, Dana},
title = {{Local Stochastic Algorithms for Alignment in SelfOrganizing Particle Systems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
pages = {14:114:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772495},
ISSN = {18688969},
year = {2022},
volume = {245},
editor = {Chakrabarti, Amit and Swamy, Chaitanya},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17136},
URN = {urn:nbn:de:0030drops171367},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.14},
annote = {Keywords: Selforganizing particle systems, alignment, Markov chains, active matter}
}
Keywords: 

Selforganizing particle systems, alignment, Markov chains, active matter 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022) 
Issue Date: 

2022 
Date of publication: 

15.09.2022 