  An infinite array of globally coupled overdamped constituents moving in a
double-well potential with $n$-th order saturation term under the influence of
additive Gaussian white noise is investigated. The system exhibits a continuous
phase transition from a symmetric phase to a symmetry-broken phase. The
qualitative behavior is independent on $n$. The critical point is calculated
for strong and for weak noise, these limits are also bounds for the critical
point. Introducing an additional nonlinearity, such that the potential can have
up to three minima, leads to richer behavior. There the parameter space divides
in three regions, a region with a symmetric phase, a region with a phase of
broken symmetry and a region where both phases coexist. The region of
coexistence collapses into one of the others via a discontinuous phase
transition whereas the transition between the symmetric phase and the phase of
broken symmetry is continuous. The tricritical point where the three regions
intersect, can be calculated for strong and for weak noise. These limiting
values form optimal bounds on the tricritical point. In the region of
coexistence simulations of finite systems are performed. One finds that the
stationary distribution of finite but large systems differs qualitatively from
the one of the infinite system. Hence the limits of stationarity and large
system size do not commute.
