Fluctuation–dissipation theorem for thermo-refractive noise

We introduce a simple prescription for calculating the spectra of thermal fluctuations of temperature-dependent quantities of the form δTˆ(t)=∫d3r→δT(r→,t)q(r→). Here T(r→,t) is the local temperature at location r→ and time t  , and q(r→) is an arbitrary function. As an example of a possible application, we compute the spectrum of thermo-refractive coating noise in LIGO, and find a complete agreement with the previous calculation of Braginsky, Gorodetsky and Vyatchanin. Our method has computational advantage, especially for non-regular or non-symmetric geometries, and for the cases where q(r→) is non-negligible in a significant fraction of the total volume.