Assume that a stochastic process can be approximated, when some
scale parameter gets large, by a fluid limit (also called mean field limit, or hydro-
dynamic limit). A common practice, often called the fixed point ap- proximation
consists in approximating the stationary behaviour of the sto- chastic process by
the stationary points of the fluid limit. It is known that this may be incorrect
in general, as the stationary behaviour of the fluid limit may not be described
by its stationary points. We show however that, if the stochastic process is
reversible, the fixed point approximation is indeed valid. More precisely, we
assume that the stochastic process converges to the fluid limit in distribution
(hence in probability) at every fixed point in time. This assumption is very
weak and holds for a large family of processes, among which many mean field
and other interaction models. We show that the reversibility of the stochastic
process implies that any limit point of its stationary distribu- tion is concen-
trated on stationary points of the fluid limit. If the fluid limit has a unique
stationary point, it is an approximation of the stationary distribution of the
stochastic process.