  We present an analysis of the stationary distributions of run-and-tumble
particles trapped in external potentials in terms of a thermophoretic
potential, that emerges when trapped active motion is mapped to trapped passive
Brownian motion in a fictitious inhomogeneous thermal bath. We elaborate on the
meaning of the non-Boltzmann-Gibbs stationary distributions that emerge as a
consequence of the persistent motion of active particles. These stationary
distributions are interpreted as a class of distributions in nonequilibrium
statistical mechanics known as superstatistics. Our analysis provides an
original insight on the link between the intrinsic nonequilibrium nature of
active motion and the well-known concept of local equilibrium used in
nonequilibrium statistical mechanics, and contributes to the understanding of
the validity of the concept of effective temperature. Particular cases of
interest, regarding specific trapping potentials used in other theoretical or
experimental studies, are discussed. We point out as an unprecedented effect,
the emergence of new modes of the stationary distribution as a consequence of
the coupling of persistent motion in a trapping potential that varies highly
enough with position.
