  The usual Langevin approach to describe systems driven by noise fails to
describe the long time behavior of systems with multiple attractors. The
solution of the associated linear Fokker-Planck equation is always unique, even
though it might show one or more maxima. In this context, it is customary to
call transitions to changes in the shape of the equilibrium distribution
function and relate the maxima to the attractors. Some years ago, a theory was
developed for a system with interacting elements or subunits that, starting
from the Langevin description of all the variables, leads to bifurcating
\textquotedblleft one-particle\textquotedblright\, distribution functions when
the number of elements tends to infinity. In this paper, a mean-field
hypothesis has been used to deal with systems with a finite number of elements.
We carry out numerical simulations yielding bifurcation solutions for the
probability density of a collective variable. We also compare the results of
the mean-field hypothesis with those obtained with the Langevin approach for
finite systems.
