Casimir energy for $N$ constant conductivity $δ$-plates with a neural network perception

Abstract: The Casimir energy for $N$ $\delta$-function plates depends on the multiple scattering parameter $\Delta$. This $N$-body interaction was distributed into interactions with nearest neighbour scattering and next-to-nearest neighbour scattering based on partitions of $N-1$ and its permutations. Implementing this methodology, we investigate the Casimir interaction for multiple plates with constant conductivity relatable to Graphene. We also study the Casimir energy between a perfect magnetic conductor and multiple constant conductivity $\delta$ plates, which results in Boyer repulsion. In the asymptotic limit for ideal boundary conditions, the results become simple where the multiple scattering parameter $\Delta$ consists only of the nearest neighbour scattering term. Further, we used neural networks to analyze the Casimir energy in the Boyer repulsion configurations to understand the influence of pairwise energies on the many-body energy. The neural network could distinguish the repulsive and attractive forces depending on the regions of varying conductivity.