  The ABC model is a simple diffusive one-dimensional non-equilibrium system
which exhibits a phase transition. Here we show that the cumulants of the
currents of particles through the system become singular near the phase
transition. At the transition, they exhibit an anomalous dependence on the
system size (an anomalous Fourier's law). An effective theory for the dynamics
of the single mode which becomes unstable at the transition allows one to
predict this anomalous scaling.

Statistical Mechanics