  We consider a two-dimensional model system of Brownian particles in which
slow particles are accelerated while fast particles are damped. The motion of
the individual particles are described by a Langevin equation with
Rayleigh-Helmholtz velocity dependent friction. In case of noninteracting
particles, the time evolution equations lead to a non-Gaussian velocity
distribution. The velocity dependent friction allows negative values of the
friction or energy intakes by slow particles which we consider as active
motion, and also causes breaking of the fluctuation dissipation relation.
Defining the effective temperature proportional to the second moment of
velocity, it is shown that for a constant effective temperature the higher the
noise strength, the lower are the number of active particles in the system.
Using the Mori-Zwanzig formalism and the mode-coupling approximation, the
equation of motion for the density auto-correlation function are derived. The
equations are solved using the equilibrium structure factors. The
integration-through-transients approach is used to derive a relation between
the structure factor in the stationary state considering the interacting
forces, and the conventional equilibrium static structure factor.
