Quantifying Gibbs measures of disordered crystals up to the solid-liquid phase transition

Abstract: Quantifying the configuration space and the Gibbs measure of thermally disordered condensed matter systems has been a long standing problem. The challenge is to avoid the Gibbs paradox, which forbids any ordering or labeling of the atoms. Our key observation is that the lattice of a thermally disordered condensed matter system, in either solid, liquid or gas phase, can be fully reconstructed from the Voronoi cells of the atoms alone, even if these Voronoi cells are disassembled and randomly scrambled. In the example of the crystalline phase of silicon, the statistics of the Voronoi cells reveals the existence of four, and only four, large facets that are present with probability one for all temperatures up to the solid-liquid melting line. These four largest facets, which separate nearest-neighboring atoms, can be also be used to reconstruct the lattice of the crystal. Hence, their collection supplies the optimal representation of the configuration of the crystal. We conjecture that the existence of Voronoi facets that, despite their large thermal fluctuations, survive with probability one up to the melting temperature, is the fundamental signature of the crystalline solid phase and therefore key to quantifying the Gibbs measure over the entire solid phase.

• The paper introduces a novel Voronoi tessellation framework to quantify Gibbs measures in thermally disordered crystals.
• The study demonstrates that persistent Voronoi cell facets reliably encode crystalline phase features up to the melting transition in silicon.
• Empirical results reveal abrupt Voronoi facet changes as markers of a first-order solid-liquid phase transition, enhancing simulation efficiency.

Quantifying Gibbs Measures of Disordered Crystals

The study conducted by Efremkin et al. addresses the longstanding challenge of characterizing Gibbs measures for thermally disordered states in condensed matter systems, with a specific focus on disordered crystals up to the solid-liquid phase transition. The primary contribution of this paper is the novel utilization of Voronoi tessellations in providing a comprehensive characterization of disordered crystalline phases.

In condensed matter physics, the concept of the Gibbs measure is employed to describe the equilibrium distribution of particle configurations at a given temperature. The existing challenge in quantifying Gibbs measures is exacerbated by thermal fluctuations and the requirement to avoid the Gibbs paradox, which is resolved by excluding labels on atoms. The authors propose a solution by leveraging the geometry of Voronoi cells, without the need to track individual atomic positions.

The paper presents empirical findings using the example of the crystalline phase of silicon. In this context, the lattice of a thermally disordered crystal can be reconstructed from an unordered set of Voronoi cells. Importantly, four specific large facets of these cells persist with a probability of one even under severe thermal fluctuations, up to the melting temperature. The study reveals that these persistent facets can encode the essential features of the crystalline phase and are key to quantifying the Gibbs measure across the entire solid phase.

The authors extend this approach to the mathematical characterization of the configuration space associated with Gibbs measures. They demonstrate that the collection of Voronoi cell geometries can represent the configuration space of the crystal. This approach allows the authors to propose a conjecture: the crystalline phase of a system is uniquely encoded by the stable facets of its Voronoi cells over the relevant temperature range.

Key numerical results from the study include the first identification of abrupt changes in Voronoi cell facets and associated statistical distributions correlated with the solid-liquid transition in silicon. These changes are indicative of a first-order phase transition. To support broad applicability, the authors assert that such Voronoi-based markers can serve as real-space indicators for crystalline phases in a diverse range of materials.

The implications of this work are multifaceted. Practically, this approach could transform the feasibility of performing first-principles simulations on disordered crystals by bypassing computationally demanding molecular dynamics simulations with expensive atomistic tracking. Theoretically, the Voronoi-based reconstruction of disordered systems offers a new lens through which researchers can view and understand the complex structural dynamics of thermally disordered systems.

The authors propose potential extensions of their methodology to encompass a wider range of crystalline materials, raising prospects for its integration into machine learning frameworks designed for material property predictions. Future research could focus on validating the conjecture across different materials and refining the Voronoi tessellation approach to encompass transitions beyond the solid-liquid boundary.

Overall, Efremkin et al. have introduced a mathematically and physically robust framework for characterizing Gibbs measures within thermally disordered crystals that may lead to new computational techniques and insights into crystalline phase behavior under thermal fluctuations.