  It has been shown that a nonvibrated magnetic granular system, when it is
feeded by means an altenating magnetic field, behaves with most of the
distinctive physical features of active matter systems. In this work we focus
our attention on the simplest granular system composed by a single magnetized
spherical particle allocated in a quasi one-dimensional circular channel that
receives energy from a magnetic field reservoir and transduces it into a
running and tumbling motion. The theoretical analysis based on the run and
tumble model on a circle of radius R forecasts the existence of a dynamical
phase transition between an erratic motion (disordered phase) when the
characteristic persistence length of the run and tumble motion, $\ell_{c} <
R/2$, to a persistent motion (ordered phase) when $\ell_{c}> R/2$. It is found
that the limiting behaviours of these phases correspond to a Brownian motion on
the circle and a simple uniform circular motion, respectively. It is
qualitatively shown that the lower magnetization of a particle, the larger
persistence lenght is. It is so at least within the experimental limit of
validity of our experiments. Our results show a very good agreement between
theory and experiment.
