  Particle motion at the micro-scale is an incessant tug-of-war between thermal
fluctuations and applied forces on one side, and the strong resistance exerted
by fluid viscosity on the other. Friction is so strong that completely
neglecting inertia - the overdamped approximation - gives an excellent
effective description of the actual particle mechanics. In sharp contrast with
this result, here we show that the overdamped approximation dramatically fails
when thermodynamic quantities such as the entropy production in the environment
is considered, in presence of temperature gradients. In the limit of
vanishingly small, yet finite inertia, we find that the entropy production is
dominated by a contribution that is anomalous, i.e. has no counterpart in the
overdamped approximation. This phenomenon, that we call entropic anomaly, is
due to a symmetry-breaking that occurs when moving to the small, finite inertia
limit. Strong production of anomalous entropy is traced back to intense sweeps
down the temperature gradient.
