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The dense packing of hard particles in confined spaces is of broad interest in both mathematics and statistical physics. It relates to classical packing problems under geometric constraints, plays a central role in understanding the self-assembly of microscopic particles such as colloids and nanoparticles, and inspires studies across a wide range of physical systems. However, achieving high packing densities under confinement remains challenging due to anisotropic particle shapes, the discontinuous nature of hard-core interactions, and geometric frustration. In this work, we develop a Monte Carlo scheme that combines boundary compression with controlled, temporary particle overlaps. Specifically, we allow a limited number of overlaps during the compression of a circular boundary, which are subsequently removed via standard Monte Carlo relaxation before further compression steps. We apply this strategy to three types of two-dimensional particles—disks, squares, and rectangles with an aspect ratio of 5:1—confined within a circular boundary. As a control, we also perform simulations using a conventional method that strictly prohibits overlaps throughout. The final configurations from both methods exhibit similar structural features. For hard disks, central particles form a triangular lattice, while those near the boundary become more disordered to accommodate the circular geometry. For hard squares, particles at the center organize into a square lattice, whereas those near the boundary form concentric layers. For rectangles, particles in the central region display local smectic-like alignment within clusters that are oriented nearly perpendicular to one another. Near the boundary, some particles align tangentially along the circular edge. Quantitatively, the temporary-overlap strategy consistently yields denser packings across all particle types. Analysis of 10 independent samples shows higher average and maximal packing densities compared to the conventional method. Further analysis of the radial distribution functions and orientational order parameters reveals that, although both methods produce similar structural features, the overlap-allowed method yields a larger central region exhibiting lattice-like or cluster-like ordering. Our findings suggest that allowing temporary particle overlaps is an effective strategy for generating dense configurations of hard particles under confinement. This approach may be extended to more complex systems, including three-dimensional particles or mixtures with different shapes confined within restricted geometries.
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Keywords:
- hard particles /
- overlap /
- packing density /
- configuration
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