  For a model long-range interacting system of classical Heisenberg spins, we
study how fluctuations, such as those arising from having a finite system size
or through interaction with the environment, affect the dynamical process of
relaxation to Boltzmann-Gibbs equilibrium. Under deterministic spin
precessional dynamics, we unveil the full range of quasistationary behavior
observed during relaxation to equilibrium, whereby the system is trapped in
nonequilibrium states for times that diverge with the system size. The
corresponding stochastic dynamics, modeling interaction with the environment
and constructed in the spirit of the stochastic Landau-Lifshitz-Gilbert
equation, however shows a fast relaxation to equilibrium on a size-independent
timescale and no signature of quasistationarity, provided the noise is strong
enough. Similar fast relaxation is also seen in Glauber Monte Carlo dynamics of
the model, thus establishing the ubiquity of what has been reported earlier in
particle dynamics (hence distinct from the spin dynamics considered here) of
long-range interacting systems, that quasistationarity observed in
deterministic dynamics is washed away by fluctuations induced through contact
with the environment.
