  We propose a theoretical framework based on $SU(3)$ coherent states as a
convenient tool to describe the collective state of a Bose-Einstein condensate
of spin 1 atoms at thermal equilibrium. We work within the single-mode
approximation, which assumes that all atoms condense in the same spatial mode.
In this system, the magnetization $m_z$ is conserved to a very good
approximation. This conservation law is included by introducing a prior
distribution for $m_z$ and constructing a generalized statistical ensemble that
preserves its first moments. In the limit of large particle numbers, we
construct the partition function at thermal equilibrium and use it to compute
various quantities of experimental interest, such as the probability
distribution function and moments of the population in each Zeeman state. When
$N$ is large but finite (as in typical experiments, where $N\sim 10^3-10^5$),
we find that fluctuations of the collective spin can be important.
