  We study the dynamics of a system of coupled oscillators of distributed
natural frequencies, by including the features of both thermal noise,
parametrized by a temperature, and inertial terms, parametrized by a moment of
inertia. For a general unimodal frequency distribution, we report here the
complete phase diagram of the model in the space of dimensionless moment of
inertia, temperature, and width of the frequency distribution. We demonstrate
that the system undergoes a nonequilibrium first-order phase transition from a
synchronized phase at low parameter values to an incoherent phase at high
values. We provide strong numerical evidence for the existence of both the
synchronized and the incoherent phase, treating the latter analytically to
obtain the corresponding linear stability threshold that bounds the first-order
transition point from below. In the limit of zero noise and inertia, when the
dynamics reduces to the one of the Kuramoto model, we recover the associated
known continuous transition. At finite noise and inertia but in the absence of
natural frequencies, the dynamics becomes that of a well-studied model of
long-range interactions, the Hamiltonian mean-field model. Close to the
first-order phase transition, we show that the escape time out of metastable
states scales exponentially with the number of oscillators, which we explain to
be stemming from the long-range nature of the interaction between the
oscillators.
