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Floquet theory is an indispensable tool for analysing periodically-drivenquantum many-body systems. Although it does not universally extend to classicalsystems, some of its methodologies can be adopted in the presence ofwell-separated timescales. Here we use these tools to investigate thestroboscopic behaviours of a classical spin chain that is driven by a periodicmagnetic field and coupled to a thermal reservoir. We detail and expand ourprevious work: we investigate the significance of higher-order corrections tothe classical Floquet-Magnus expansion in both the high- and low-frequencyregimes; explicitly probe the evolution the dynamics of the reservoir; andfurther explore how the driven system synchronises with the applied field atlow frequencies. In line with our earlier results, we find that thehigh-frequency regime is characterised by a local Floquet-Gibbs ensemble withthe reservoir acting as a nearly-reversible heatsink. At low frequencies, thedriven system rapidly enters a synchronised state, which can only be fullydescribed in a global picture accounting for the concurrent relaxation of thereservoir in a fictitious magnetic field arising from the drive. We highlighthow the evolving nature of the reservoir may still be incorporated in a localpicture by introducing an effective temperature. Finally, we argue thatdissipative equations of motion for periodically-driven many-body systems, atleast at intermediate frequencies, must generically be non-Markovian.

Reservoir-induced stabilisation of a periodically driven classical spin chain: local vs. global relaxation