Temperature transitions and degeneracy in the control of small clusters with a macroscopic field

TL;DR

This paper numerically investigates how to optimally control small fluctuating clusters with a macroscopic field to reach a target shape quickly, revealing temperature-dependent policy transitions and symmetry effects.

Contribution

It introduces a dynamic programming approach to determine optimal control policies for cluster shape manipulation, highlighting degeneracy and temperature transition phenomena.

Findings

Optimal control policies are non-unique due to symmetries.

The fraction of shapes with non-unique force choices decreases with cluster size.

Temperature variations cause discrete transitions in the control policy.

Abstract

We present a numerical investigation of the control of few-particle fluctuating clusters with a macroscopic field. Our goal is to reach a given target cluster shape is minimum time. This question is formulated as a first passage problem in the space of cluster configurations. We find the optimal policy to set the macroscopic field as a function of the observed shape using dynamic programming. Our results show that the optimal policy is non-unique, and its degeneracy is mainly related to symmetries shared by the initial shape, the force and the target shape. The total fraction of shapes for which optimal choice of the force is non-unique vanishes as the cluster size increases. Furthermore, the optimal policy exhibits a discrete set of transitions when the temperature is varied. Each transition leads to a discontinuity in the derivative of the time to reach with target with respect to temperature. As the size of the cluster increases, the change in the policy due to temperature transitions grows like the total number of configurations and a continuum limit emerges.