Emergence of zero modes in disordered solids under periodic tiling.

Area of Science:

• Computational physics
• Materials science
• Statistical mechanics

Background:

• Periodic boundary conditions (PBC) are commonly used in computational models to simulate bulk materials.
• PBC assume a system is identical to infinite copies of itself, constraining particle movements across boundaries.
• This assumption may not hold for all material structures, especially jammed packings or elastic networks.

Purpose of the Study:

• To investigate the validity of periodic boundary conditions in computational models of particle packings.
• To determine if a jammed packing under PBC accurately reflects the properties of its infinite lattice representation.
• To identify the limitations and successes of PBC in capturing the physics of repeated structures.

Main Methods:

• Theoretical analysis of particle packing models.
• Mathematical proof of the non-rigidity of infinite lattice representations derived from PBC simulations.
• Comparison of properties between PBC-simulated packings and their infinite counterparts.

Main Results:

• A jammed packing or rigid elastic network simulated with PBC may not correspond to a rigid or energy-minimized infinite lattice.
• The constraint of moving particle images together in PBC can lead to artificial rigidity or stability.
• Infinite lattice representations may exhibit different mechanical and energetic properties than their PBC-simulated counterparts.

Conclusions:

• Periodic boundary conditions can be a useful approximation but may fail to capture the true physics of infinite systems.
• The study highlights critical discrepancies between PBC simulations and actual infinite structures, particularly for jammed states.
• Understanding these limitations is crucial for accurate computational modeling in materials science and related fields.