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We study long-range interacting systems driven by external stochastic forcesthat act collectively on all the particles constituting the system. Such ascenario is frequently encountered in the context of plasmas, self-gravitatingsystems, two-dimensional turbulence, and also in a broad class of othersystems. Under the effect of stochastic driving, the system reaches astationary state where external forces balance dissipation on average. Thesestates have the invariant probability that does not respect detailed balance,and are characterized by non-vanishing currents of conserved quantities. Inorder to analyze spatially homogeneous stationary states, we develop a kineticapproach that generalizes the one known for deterministic long-range systems;we obtain a very good agreement between predictions from kinetic theory andextensive numerical simulations. Our approach may also be generalized todescribe spatially inhomogeneous stationary states. We also report on numericalsimulations exhibiting a first-order nonequilibrium phase transition fromhomogeneous to inhomogeneous states. Close to the phase transition, the systemshows bistable behavior between the two states, with a mean residence time thatdiverges as an exponential in the inverse of the strength of the externalstochastic forces, in the limit of low values of such forces.

Kinetic theory of nonequilibrium stochastic long-range systems: Phase transition and bistability