A statistical theory of structure in many-particle systems with local interactions

TL;DR

The paper develops a rigorous statistical framework for structure in many-particle systems with local interactions, showing that equilibrium structure is fully determined by a field of fine local descriptors and, when these are coarse-grained, can be approximated via a local random field. It defines a universal local order quantifier based on angular redundancy (extracopularity) and links order to symmetry through a sharp point-group bound, with equality in highly symmetric geometries. The Extracopularity framework yields a closed-form, Gaussian-mixture distribution (XDF) for highly coordinated states and is applied to the ideal gas, perfect crystals, and simple liquids, producing explicit models for coordination, bond-angle entropy, and temperature/gradient effects. Collectively, the results provide a tractable, invariant-descriptor approach to comparing and understanding structure across crystalline, liquid, and gaseous phases, while enabling coarse-graining and nonequilibrium extensions. The work thereby offers a principled bridge between local geometric descriptors and global structural order with potential for broad applicability in materials science and condensed-matter theory.

Abstract

A theory of structure is formulated for systems of many structureless classical particles with stable local interactions in Euclidean space. Such systems are shown to have their structure in thermodynamic equilibrium determined exactly by a random field of fine local descriptions and approximately by coarsenings thereof. The degree of order in the local cluster consisting of a particle and its neighbors is identified as a universal source of coarse local descriptions and characterized by expressing the behavior of configurational entropy in local microscopic terms. A local measure of the angular redundancy in neighboring particle positions is found to satisfy this characterization and thereby established as a valid local order quantifier. A precise relationship between order and symmetry is obtained by bounding this quantifier sharply from below by a simple function of the local point group and the largest stabilizer under its action on the set of bond pairs. The marginal distribution of the quantifier is given in closed form for highly coordinated particles with broadly distributed bond angles. Applications are made to the ideal gas, perfect crystal, and simple liquid.

Figures (11)

• Figure 1: Radius-$1$ graphs include (a) star graphs and (b) complete graphs, but also wheel graphs, fan graphs, and others.
• Figure 2: Three classes of structural descriptions: (a) labeled pairwise distances (fine local); (b) labeled neighbor vectors (fine nonlocal); (c) inter-neighbor vectors (non-fine local).
• Figure 3: Absolute net magnetization per site in a zero-field Ising ferromagnet: In its ground state, the system assumes configuration (a) or its inversion (b) with probability one, giving $\left\lvert M\right\rvert/N = 1$ (maximum order). Above the Curie point, each spin points in either direction with equal probability, giving $\langle \left\lvert M\right\rvert/N \rangle = 0$ as typified in (c), with mutually independent spins as $T \to \infty$ (complete randomness).
• Figure 4: A regular pentagonal bond set (center) and the partition of its bond pairs by angle: $72^\circ$ (right) and $144^\circ$ (left).
• Figure 5: The stabilizer of the bond pair $\{b_1,b_2\}$ under the action of the tetrahedral point group $T_d$: (a) the identity $e$, (b) the rotation $R$, (c) the reflection $M$, and (b) the reflection $M'$. Here bonds are represented not by arrows as is usually done but by the vertices at their terminal points, so that bond pairs correspond to the line segments joining those vertices.

Theorems & Definitions (15)

• Definition 3: local order quantifier
• Remark 4
• Lemma 5
• proof
• Theorem 6
• proof
• Corollary 7
• proof
• Example 8
• Definition 9: spherical