When the additivity principle fails for current fluctuations
In plain words
Large current fluctuations in a driven conductor can be computed from a principle that assumes the most likely fluctuation keeps a fixed density profile in time. For some systems and currents the most likely fluctuation instead changes its profile in time or breaks a symmetry of the system, which is a phase transition in the fluctuations.
Precise statement
For one-dimensional boundary-driven diffusive systems described by macroscopic fluctuation theory with diffusivity $D(\rho)$ and mobility $\sigma(\rho)$, classify the ($D, \sigma$, boundary densities, current $j$) for which the large deviation function of the time-integrated current is given by the time-independent additivity-principle solution, and where time-dependent optimal paths (traveling waves on a ring) or symmetry-broken time-independent profiles take over. An answer is a necessary and sufficient criterion.
What would settle it
A proof of a necessary and sufficient condition within macroscopic fluctuation theory, checked against exact results for the KMP and exclusion models.
Status in the literature
Sufficient conditions for stability of the additivity solution and explicit violations are known, including symmetry-breaking transitions in boundary-driven systems (Baek, Kafri, Lecomte, Phys. Rev. Lett. 118, 030604, 2017); a general criterion is not.