Momentum coupling of classical atoms

TL;DR

This work introduces a dual-space cluster expansion for classical particles in the continuum, translating configuration-space variables into momenta of relative displacements through a Fourier expansion. By enforcing cyclic validity for clusters, it derives a recurrence relation for the partition function and constructs a thermodynamic-limit framework where finite clusters carry density up to a critical threshold $\rho_c(β)$; beyond this threshold, infinite clusters must appear, signaling a phase-transition-like singularity. An illustrative Gaussian example shows a finite $\rho_c$ in $d\ge 2$, and the analysis argues that all condensed phases involve infinite clusters, with submacroscopic clusters expected in liquids and macroscopic ones in crystals. Overall, the paper provides a new analytical lens linking continuum phase transitions to percolation-like cluster formation via momentum-space variables and cyclic cluster constraints.

Abstract

A new method, dual-space cluster expansion, is proposed to study classical phases transitions in the continuum. It relies on replacing the particle positions as integration variables by the momenta of the relative displacements of particle pairs. Due to the requirement that the particles must be static, coupling via the momenta partitions the set of particles into a set of clusters, and transforms the partition function into a sum over the different cluster decompositions. This allows us to derive a formula for the density that finite clusters can carry in the infinite system. In a simplified example, we then demonstrate that in two and higher dimensions this density has a threshold, beyond which the particles form infinite clusters. The transition is accompanied by a singularity in the free energy. We also show that infinite clusters are always present in condensed phases, most likely submacroscopic in liquids and macroscopic in crystals.