  We study the statistical distribution of the ergotropy and of the efficiency
of a single-qubit battery ad of a single-qubit Otto engine, respectively
fuelled by random collisions. The single qubit, our working fluid, is assumed
to exchange energy with two reservoirs, a non-equilibrium "hot" reservoir and a
zero temperature cold reservoir. The interactions between the qubit and the
reservoirs is described in terms of a collision model of open system dynamics.
The qubit interacts with the non-equilibrium reservoir (a large ensemble of
qudits all prepared in the same pure state) via random unitary collisions and
with the cold reservoir (a large ensemble of qubits in their ground state) via
a partial swap. Due to the random nature of the interaction with the hot
reservoir, fluctuations in ergotropy, heat, and work are present, shrinking
with the size of the qudits in the hot reservoir. While the mean, "macroscopic"
efficiency of the Otto engine is the same as in the case in which the hot
reservoir is a thermal one, the distribution of efficiencies does not support
finite moments, so that the mean of efficiencies does not coincide with the
macroscopic efficiency.
