Random Sequential Addition of Hard Spheres in High Euclidean Dimensions

TL;DR

This study combines numerical and theoretical approaches to analyze the structure of randomly added spheres in high-dimensional spaces, revealing how saturation density and correlations change with dimension, with implications for disordered ground states.

Contribution

It provides the first detailed analysis of RSA sphere packings in dimensions 1 through 6, including new scaling laws and bounds for saturation density in high dimensions.

Findings

Saturation density scales as c1/2^d + c2 d/2^d for 2 ≤ d ≤ 6.

Pair correlations diminish significantly as dimension increases.

A lower bound on saturation density is derived based on the structure factor.

Abstract

Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in $d$-dimensional Euclidean space $\mathbb{R}^d$ in the infinite-time or saturation limit for the first six space dimensions ($1 \le d \le 6$). Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each =of these dimensions. We find that for $2 \le d \le 6$, the saturation density $\phi_s$ scales with dimension as $\phi_s= c_1/2^d+c_2 d/2^d$, where $c_1=0.202048$ and $c_2=0.973872$. We also show analytically that the same density scaling persists in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any $d$ given by $\phi_s \ge (d+2)(1-S_0)/2^{d+1}$, where $S_0\in [0,1]$ is the structure factor at $k=0$ (i.e., infinite-wavelength number variance) in the high-dimensional limit. Consistent with the recent "decorrelation principle," we find that pair correlations markedly diminish as the space dimension increases up to six. Our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions.