  The test mass suspensions of cryogenic gravitational-wave detectors such as
the KAGRA project are tasked with extracting the heat deposited on the optics.
Thus these suspensions have a non-uniform temperature, requiring the
calculation of thermal noise in non-equilibrium conditions. While it is not
possible to describe the whole suspension system with one temperature, the
local temperature anywhere in the system is still well defined. We therefore
generalize the application of the fluctuation-dissipation theorem to mechanical
systems, pioneered by Saulson and Levin, to non-equilibrium conditions in which
a temperature can only be defined locally. The result is intuitive in the sense
that the temperature-averaging relevant for the thermal noise in the observed
degree of freedom is given by averaging the temperature field, weighted by the
dissipation density associated with that particular degree of freedom. After
proving this theorem we apply the result to examples of increasing complexity:
a simple spring, the bending of a pendulum suspension fiber, as well as a model
of the KAGRA cryogenic suspension. We conclude by outlining the application to
non-equilibrium thermo-elastic noise.
