  We consider diffusive lattice gases on a ring and analyze the stability of
their density profiles conditionally to a current deviation. Depending on the
current, one observes a phase transition between a regime where the density
remains constant and another regime where the density becomes time dependent.
Numerical data confirm this phase transition. This time dependent profile
persists in the large drift limit and allows one to understand on physical
grounds the results obtained earlier for the totally asymmetric exclusion
process on a ring.
