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Classical Heisenberg spins with long-range interactions: Relaxation to equilibrium for finite systems

Systems with long-range interactions often relax towards statistical equilibrium over timescales that diverge with $N$, the number of particles. A recent work [S. Gupta and D. Mukamel, J. Stat. Mech.: Theory Exp. P03015 (2011)] analyzed a model system comprising $N$ globally coupled classical Heisenberg spins and evolving under classical spin dynamics. It was numerically shown to relax to equilibrium over a time that scales superlinearly with $N$. Here, we present a detailed study of the Lenard-Balescu operator that accounts at leading order for the finite-$N$ effects driving this relaxation. We demonstrate that corrections at this order are identically zero, so that relaxation occurs over a time longer than of order $N$, in agreement with the reported numerical results.