Direct approach for the fluctuation-dissipation theorem under nonequilibrium steady-state conditions

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.