The basins of attraction of soft-sphere packings are not fractal

Abstract

The energy landscape picture is a central tool to study many-body systems. In particular, the energy landscapes of glass-forming liquids, jammed packings, constraint satisfaction problems, or neural networks contain a plethora of minima corresponding to competing states. Due to their complexity, these landscapes resist analytical treatment and must be studied numerically. We focus on jammed soft spheres, a paradigmatic model of glasses and granulars, to expose the limitations of standard numerical methods in resolving the true structure of energy landscapes. We show that the C-language Variable-coefficients ordinary differential equation solver is the one with the best time-for-error trade-off for this problem, outperforming commonly used steepest-descent solvers by several orders of magnitude. Using this numerical approach, we provide unequivocal evidence that optimizers widely used in computational studies destroy all semblance of the true landscape geometry, even in moderately low dimensions. Employing a range of geometric indicators, both low- and high-dimensional, we show that earlier claims on the fractality of basins of attraction of minima originated from the use of inadequate mapping strategies. In reality, the basins of attraction of soft-sphere packings are smooth structures with well-defined length scales, a result that likely extends to a much broader family of problems.

The authors' abstract, as published at the source. Proceedings of the National Academy of Sciences, 2026 · DOI ↗