  The out-of-equilibrium dynamics of the Hamiltonian Mean Field (HMF) model is
studied in presence of an externally imposed magnetic field h. Lynden-Bell's
theory of violent relaxation is revisited and shown to adequately capture the
system dynamics, as revealed by direct Vlasov based numerical simulations in
the limit of vanishing field. This includes the existence of an
out-of-equilibrium phase transition separating magnetized and non magnetized
phases. We also monitor the fluctuations in time of the magnetization, which
allows us to elaborate on the choice of the correct order parameter when
challenging the performance of Lynden-Bell's theory. The presence of the field
h removes the phase transition, as it happens at equilibrium. Moreover, regions
with negative susceptibility are numerically found to occur, in agreement with
the predictions of the theory.
