  We consider a paradigmatic model describing the one-dimensional motion of $N$
rotators coupled through a mean-field interaction, and subject to the
perturbation of an external magnetic field. The latter is shown to
significantly alter the system behaviour, driving the emergence of ensemble
inequivalence in the out-of-equilibrium phase, as signalled by a negative
(microcanonical) magnetic susceptibility. The thermodynamic of the system is
analytically discussed, building on a maximum entropy scheme justified from
first principles. Simulations confirm the adequacy of the theoretical picture.
Ensemble inequivalence is shown to rely on a peculiar phenomenon, different
from the one observed in previous works. As a result, the existence of a convex
intruder in the micro-canonical energy is found to be a necessary but not
sufficient condition for inequivalence to be (macroscopically) observed.
Negative temperature states are also found to occur. These intriguing phenomena
reflect the non-Boltzmanian nature of the scrutinized problem and, as such,
bear traits of universality that embraces equilibrium as well
out-of-equilibrium regimes.
