Configurational-space separation and structure selection in three hard squares
Abstract
Self-assembly of hard particles with diverse shapes gives rise to a rich variety of structures through excluded-volume constraints alone. Here we show that even a minimal system of three hard squares confined in a two-dimensional periodic box exhibits nontrivial configurational behavior relevant to structure selection. As the packing fraction increases, radial distribution functions obtained from Markov-chain Monte Carlo and uniform non-overlapping insertion sampling agree at low densities, deviate markedly over an intermediate range, and converge again at higher densities. Pressure measurements provide strong numerical evidence that the discrepancy originates from the separation of the allowed configurational space into two disconnected regions above a characteristic density. We identify the separation density as ϕsep=3/5, construct explicit overlap-free transition pathways connecting the two regions immediately below it, and quantify their relative configurational-space volumes. At higher packing fractions, an approximately L-shaped arrangement of the particle centers becomes strongly favored over a staggered one, revealing a structural motif characteristic of tetratic and square-lattice ordering in larger hard-square systems. These results show that excluded-volume geometry can govern both configurational connectivity and local structure selection even in a three-particle system, revealing how signatures of many-particle self-assembly can already emerge in the few-particle limit.
Related
- When Cooling Meets Flow: Advection-Driven Orientation Selection in Phase Separation around a Moving Cooling Source
- Doubling down on proton structure: what double parton scattering can and cannot tell us about partonic separation
- Rethinking free-space orbital angular momentum quantum key distribution: encoding selection and branch-cut-length-minimized adaptive optics
- Real-space construction and selection rules of center-of-mass momentum eigenstates in two-dimensional many-body lattices
- Real-Space Selection Rule for Moiré Exciton-Phonon Coupling
BibTeX
@misc{yang2026configurationalspace,
title = {{Configurational-space separation and structure selection in three hard squares}},
author = {Yuheng Yang and Meng Xiao and Duanduan Wan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09936},
eprint = {2609.09936},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09936},
}
RIS
TY - UNPB
TI - Configurational-space separation and structure selection in three hard squares
AU - Yuheng Yang
AU - Meng Xiao
AU - Duanduan Wan
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09936
UR - https://arxiv.org/abs/2609.09936
AB - Self-assembly of hard particles with diverse shapes gives rise to a rich variety of structures through excluded-volume constraints alone. Here we show that even a minimal system of three hard squares confined in a two-dimensional periodic box exhibits nontrivial configurational behavior relevant to structure selection. As the packing fraction increases, radial distribution functions obtained from Markov-chain Monte Carlo and uniform non-overlapping insertion sampling agree at low densities, deviate markedly over an intermediate range, and converge again at higher densities. Pressure measurements provide strong numerical evidence that the discrepancy originates from the separation of the allowed configurational space into two disconnected regions above a characteristic density. We identify the separation density as ϕ_sep=3/5, construct explicit overlap-free transition pathways connecting the two regions immediately below it, and quantify their relative configurational-space volumes. At higher packing fractions, an approximately L-shaped arrangement of the particle centers becomes strongly favored over a staggered one, revealing a structural motif characteristic of tetratic and square-lattice ordering in larger hard-square systems. These results show that excluded-volume geometry can govern both configurational connectivity and local structure selection even in a three-particle system, revealing how signatures of many-particle self-assembly can already emerge in the few-particle limit.
ER -