  We are used to measure temperature with a thermometer and we know from
everyday life that different types of thermometers measure the same
temperature. This experience can be based on equilibrium thermodynamics, which
explains the equivalence of different possibilities to define temperature. In
contrast, for systems out of equilibrium such as active matter, measurements
performed with different thermometers can generally lead to different
temperature values. In the present work, we systematically compare different
possibilities to define temperature for active systems. Based on simulations
and theory for inertial active Brownian particles, we find that different
temperatures generally lead to different temperature values, as expected.
Remarkably, however, we find that different temperatures not only lead to the
same values near equilibrium (low P\'eclet number or high particle mass), but
even far from equilibrium and in cases where entropy production is large,
several different temperatures approximately coincide. In particular, we find
that the kinetic temperature, the configurational temperature, and temperatures
based on higher moments of the velocity distribution constitute a class of
temperatures that all assume very similar values over a wide parameter range.
The effective temperature also leads to similar temperature values but only in
a limited parameter regime. Notably, temperatures exploiting the virial
theorem, the Stokes-Einstein relation, or a harmonic confinement form a second
class of temperatures whose values approximately coincide with each other but
which strongly differ from those of the first class. Finally, we identify
advantages and disadvantages of the different possibilities to define
temperature and discuss their relevance for measuring the temperature of an
active system.
