Effective dynamics from a multi-scale, symmetry-preserving perturbation theory and its applications

We provide a useful method of deriving the slow, effective dynamics of an autonomous differential equation that preserves some desired properties of the original system. By an appropriate transformation from the original system into a simpler effective system by means of a Magnus exponential, the group symmetries are preserved independent of the truncation order. The formalism relies on a simple and intuitive approach to address fast and slow timescales common within experiments that allows for higher-order corrections to the effective dynamics from the fast, high-frequency terms. The formalism was previously introduced in the context of quantum information science as quantum averaging theory (QAT) but can be formally generalized to any autonomous system allowing for a wide range of applications. We will demonstrate the exciting capabilities of the QAT method with examples from quantum information science and discuss how this technique can be used to develop effective models of multi-scale plasma dynamics.