The Basins of Attraction of Soft Sphere Packings Are Not Fractal

The energy landscape plays a central role in understanding the dynamics of low-temperature and athermal systems, which are characterized by a plethora of minima and the intricate geometry of their basins of attraction. Due to their complexity, these landscapes resist analytical treatment and must instead 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 CVODE is the ODE solver with the best time-for-error trade-off, 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 of soft sphere packings 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 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.