  The question of characterization of the degree of non-equilibrium activity in
active matter systems is studied in the context of a stochastic microswimmer
model driven by a chemical cycle. The resulting dynamical properties and
entropy production rate unravel a complex interplay between the chemical and
the hydrodynamic degrees of freedom beyond linear response, which is not
captured by conventional phenomenological approaches. By studying the
precision-dissipation trade-off, a new protocol is proposed in which
microscopic chemical driving forces can be inferred experimentally. Our
findings highlight subtleties associated with the stochastic thermodynamics of
autonomous microswimmers.
