  We consider the thermal and athermal overdamped motion of particles in 1D
geometries where discrete internal degrees of freedom (spin) are coupled with
the translational motion. Adding a driving velocity that depends on the
time-dependent spin constitutes the simplest model of active particles
(run-and-tumble processes) where the violation of the equipartition principle
and of the Sutherland-Einstein relation can be studied in detail even when
there is generalized reversibility. We give an example (with four spin values)
where the irreversibility of the translational motion manifests itself only in
higher-order (than two) time correlations. We derive a generalized telegraph
equation as the Smoluchowski equation for the spatial density for an arbitrary
number of spin values. We also investigate the Arrhenius exponential law for
run-and-tumble particles, due to their activity the slope of the potential
becomes important in contrast to the passive diffusion case and activity
enhances the escape from a potential well (if that slope is high enough).
Finally, in the absence of a driving velocity, the presence of internal
currents such as in the chemistry of molecular motors may be transmitted to the
translational motion and the internal activity is crucial for the direction of
the emerging spatial current.
