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 (Gupta and Mukamel 2011 J. Stat. Mech. P03015) 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.