  The presence of defects in solids formed by active particles breaks their
discrete translational symmetry. As a consequence, many of their properties
become space-dependent and different from those characterizing perfectly
ordered structures. Motivated by recent numerical investigations concerning the
nonuniform distribution of entropy production and its relation to the
configurational properties of active systems, we study theoretically and
numerically the spatial profile of the entropy production rate when an active
solid contains an isotopic mass defect. The theoretical study of such an
imperfect active crystal is conducted by employing a perturbative analysis that
considers the perfectly ordered harmonic solid as a reference system. The
perturbation theory predicts a nonuniform profile of the entropy production
extending over large distances from the position of the impurity. The entropy
production rate decays exponentially to its bulk value with a typical healing
length that coincides with the correlation length of the spatial velocity
correlations characterizing the perfect active solids in the absence of
impurities. The theory is validated against numerical simulations of an active
Brownian particle crystal in two dimensions with Weeks-Chandler-Andersen
repulsive interparticle potential.
