Numerically discovered inherent states are always protocol dependent in jammed packings

The energy landscape for soft sphere packings exists in a high-dimensional space and plays host to an astronomical number of local minima in a hierarchical or ultrametric arrangement. From any given starting configuration we can unambiguously identify the "inherent state" as the local minimum to which the configuration will flow under perfectly damped dynamics (i.e. velocity is proportional to force). These continuous dynamics can be approximated by various discrete numerical techniques, such as steepest-descent dynamics, which are parameterized in some form by an effective step-size. Can discrete numerical minimizers always find the same minima as the zero step-size perfectly damped dynamics? Here, we measure the largest step-size parameter for a given starting configuration, such that every smaller step-size, down to the smallest value we can probe, minimizes to the same location using a steepest descent minimizer. We find that there exists a "typical" step-size that works for most starting configurations and thus would be considered "good enough" from a rough sampling. However, there always exist configurations for which the minima is dependent on the step-size, down to the smallest step-size we probe. We can understand this dependence as the result of an effective momentum introduced to our minimization through the finite step size. This momentum is able to carry the system across borders between inherent states that are found at low energy saddle points, which are nearly entirely attractive in high dimensional spaces. Thus, there is no universal "best" step-size for steepest descent dynamics which guarantees minimization to the inherent state. Therefore, we cannot unambiguously numerically determine inherent states in this energy landscape.