We introduce a simple prescription for calculating the spectra of thermal fluctuations of temperature-dependent quantities of the form $\hat{ \delta T}(t)=\int d^3\vec{ r} \delta T(\vec{ r},t) q(\vec{ r})$. Here $T(\vec{ r}, t)$ is the local temperature at location $\vec{ r}$ and time $t$, and $q(\vec{ r})$ 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 $q(\vec{ r})$ is non-negligible in a significant fraction of the total volume.
Categories: cond-mat.stat-mech astro-ph gr-qc