  We develop the kinetic theory of the flux-carrying Brownian motion recently
introduced in the context of open quantum systems. This model constitutes an
effective description of two-dimensional dissipative particles violating both
time-reversal and parity that is consistent with standard thermodynamics. By
making use of an appropriate Breit-Wigner approximation, we derive the general
form of its quantum kinetic equation for weak system-environment coupling. This
encompasses the well-known Kramers equation of conventional Brownian motion as
a particular instance. The influence of the underlying chiral symmetry is
essentially twofold: the anomalous diffusive tensor picks up antisymmtretic
components, and the drift term has an additional contribution which plays the
role of an environmental torque acting upon the system particles. These yield
an unconventional fluid dynamics that is absent in the standard
(two-dimensional) Brownian motion subject to an external magnetic field or an
active torque. For instance, the quantum single-particle system displays a
dissipationless vortex flow in sharp contrast with ordinary diffusive fluids.
We also provide preliminary results concerning the relevant hydrodynamics
quantities, including the fluid vorticity and the vorticity flux, for the
dilute scenario near thermal equilibrium. In particular, the flux-carrying
effects manifest as vorticity sources in the Kelvin's circulation equation.
Conversely, the energy kinetic density remains unchanged and the usual Boyle's
law is recovered up to a reformulation of the kinetic temperature.
