  We investigate the effect of coarse-graining on the energetics properties of
a system, focusing on entropy production. As a case of study, we consider a
one-dimensional colloidal particle in contact with a thermal bath, moving in a
sinusoidal potential and driven out of equilibrium by a small constant force.
Different levels of coarse-graining are evaluated: at first, we compare the
results in the underdamped dynamics with those in the overdamped one (first
coarse-graining). For large values of the viscosity, the two dynamics have the
same energetic properties, while, for smaller viscosity values, the overdamped
approximation produces an excess of entropy production with respect to that of
the underdamped dynamics. Moreover, for further smaller values of the drag
coefficient, the excess of entropy production turns into a loss. These regimes
are explained by evaluating the jump statistics, observing that the inertia is
able to induce multiple jumps and affect the average jump rate. The periodic
shape of the potential allows us to approximate the continuous dynamics via a
Markov chain, after the introduction of a suitable time and space
discretization (second level of coarse-graining). This discretization procedure
is implemented starting both from the underdamped and the overdamped evolution
and is analyzed for different values of the viscosity.
