## What is/are Double Sphere?

Double Sphere - Measurements are presented for a double sphere consisting of two golf balls joined together, incident obliquely on a smooth horizontal surface.^{[1]}For the generalized Lohe matrix model corresponding to the Lohe tensor model for rank-2 tensors with the same size, we observe that the component rank-1 tensors of the completely separable states satisfy the swarm double sphere model introduced in [Lohe in Physica D 412, 2020].

^{[2]}The single sphere model in MD simulations was studied for typical adsorption behaviors, and the double sphere model was built to be more focused on the gap impact between two particles.

^{[3]}We have developed an unsupervised machine learning algorithm based on Gaussian mixture model (GMM) to classify 1965 particles made of single and double spheres, cylinders and ellipsoids with varied RI values in parameter space.

^{[4]}We focus our study on a system formed by a double sphere core-shell, represented by the Ising model.

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## double sphere model

For the generalized Lohe matrix model corresponding to the Lohe tensor model for rank-2 tensors with the same size, we observe that the component rank-1 tensors of the completely separable states satisfy the swarm double sphere model introduced in [Lohe in Physica D 412, 2020].^{[1]}The single sphere model in MD simulations was studied for typical adsorption behaviors, and the double sphere model was built to be more focused on the gap impact between two particles.

^{[2]}