Random close packing at extreme size ratios with an Adam-based inflation protocol

Abstract

We present \texttt{rcpgenerator}, an openly available code for generating $d$-dimensional dense, disordered, non-overlapping close packings from an arbitrary prescribed list of particle diameters. The method adapts the Clarke--Wiley inflation protocol, but instead uses the Adam optimizer to relax the particle configuration. Typically, particle coordinates are advanced with a single, global step size, which must shrink as the size ratio $S\equiv D_{\max }/D_{\min }$ grows, generally stalling the optimization. Adam instead gives each coordinate its own adaptive step size, stabilizing the optimization time across a broader range of $S$. We demonstrate this in three-dimensional periodic tests that reach $S\sim 5\times 10^5$ for a continuous lognormal distribution ($N\sim 10^6$ diameters) and particle numbers up to $N\approx 5.6\times 10^6$ for power-law distributions, with the densest packings reaching $\varphi \simeq 0.87$, each completed in minutes to hours on a multicore machine. Across truncated-lognormal, truncated-power-law, and Weibull distributions, the resulting $\varphi$ reproduces trends such as the locations of peaks and knees with distribution shape and $S$ found in prior numerical results and in the parameter-free Farr--Groot prediction, with a remaining offset typically $0.005$--$0.01$. Additionally, results are commensurate with multimodal packing densities measured in vibrated-bed experiments. The code and the complete per-case census behind every figure are released with the paper.

11 pages, 10 figures, 3 tables, Code: https://github.com/KD-physics/RCPGenerator; data archived at Zenodo, doi:10.5281/zenodo.21435447