Magnetic materials are commonly analysed using the Landau--Lifshitz--Gilbert (LLG) equation. Despite wide use this equation contains a phenomenological damping term that remains poorly understood and extensions beyond the LLG equation have been plagued with having to put the fluctuation-dissipation-relation (FDR) in by hand. I will discuss how the a system+bath Hamiltonian, similar to the Caldeira-Leggett and spin-boson models, can be used to derive a general spin dynamics equation that automatically fulfils the FDR, and show that the new equation reduces to the standard LLG equation for Ohmic coupling.
I will demonstrate how resonant Lorentzian couplings can be used as a tool for the systematic comparison of spin dynamics in LLG-like and non-LLG-like regimes, and present numerical results showing their magnetisation dynamics and steady state. We find much quicker decay to steady state for couplings that invoke memory effects, and quantum flattening of the magnetization curve of a single classical spin at low temperatures. The presented model provides a powerful tool to explore general three-dimensional rotation and dissipation in quantum and classical thermodynamics.
 arxiv 2009.00600v1 (2020), J.Anders, C.R.J. Sait, S.A.R. Horsley.
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