  We derive a master equation for the reduced density matrix of a uniformly
accelerating quantum detector in arbitrary dimensions, generically coupled to a
field initially in its vacuum state, and analyze its late time regime. We find
that such density matrix asymptotically reaches a Gibbs state. The
particularities of its evolution towards this state are encoded in the response
function, which depends on the dimension, the properties of the fields, and the
specific coupling to them. We also compare this situation with the
thermalization of a static detector immersed in a thermal field state,
pinpointing the differences between both scenarios. In particular, we analyze
the role of the response function and its effect on the evolution of the
detector towards equilibrium. Furthermore, we explore the consequences of the
well-known statistics inversion of the response function of an Unruh-DeWitt
detector linearly coupled to a free scalar field in odd spacetime dimensions.
This allows us to specify in which sense accelerated detectors in Minkowski
vacuum behave as static detectors in a thermal bath and in which sense they do
not.
