Kinetic description of one-dimensional stochastic dynamics with small inertia

We study single-variable approaches for describing stochastic dynamics with small inertia. The basic models we deal with describe passive Brownian particles and phase elements (phase oscillators, rotators, superconducting Josephson junctions) with an effective inertia in the case of a linear dissipation term and active Brownian particles in the case of a nonlinear dissipation. Elimination of a fast variable (velocity) reduces the characterization of the system state to a single variable and is formulated in four representations: moments, cumulants, the basis of Hermite functions, and the formal cumulant variant of the last. This elimination provides rigorous mathematical description for the overdamped limit in the case of linear dissipation and the overactive limit of active Brownian particles. For the former, we derive a low-dimensional equation system which generalizes the Ott-Antonsen Ansatz to systems with small effective inertia. In the latter case, we derive a Fokker-Planck-type equation with a forced drift term and an effective diffusion in one dimension, where the standard two- and three-dimensional mechanism is impossible. In the four considered representations, truncated equation chains are demonstrated to be utilitary for numerical simulation for a small finite inertia.