Theory of Generalized Hertzian Hyperspheres

TL;DR

This paper develops a theoretical framework and numerical validation for generalized Hertzian hyperspheres, a class of nearly hard-sphere systems with soft, finite-range repulsive interactions, across multiple dimensions.

Contribution

It introduces a unified theory for soft-sphere systems with finite-range repulsions, deriving explicit thermodynamic and scaling laws, and validates these predictions through extensive simulations from 1D to 8D.

Findings

Derived closed-form thermodynamic expressions.

Established scaling laws for structure and dynamics.

Validated predictions through multi-dimensional simulations.

Abstract

While hard-sphere models form the foundation of theoretical condensed matter physics, real systems often exhibit some degree of softness. We present a theoretical and numerical study of a class of nearly hard-sphere systems, generalized Hertzian hyperspheres, where particles interact via a finite-range repulsive potential that allows slight overlaps. Well-studied examples of this class include particles with harmonic repulsions, Hertzian spheres, and Hertzian disks. We derive closed-form expressions for thermodynamic properties, coexistence pressures, and scaling laws governing structure and dynamics. The theory predicts how quantities scale with temperature, density, spatial dimension, and potential softness. These theoretical predictions are tested through numerical simulations in dimensions ranging from one to eight.