Heating Through Phonon Excitation Implied by Collapse Models

We calculate the rate of heating through phonon excitation implied by the noise postulated in mass-proportional-coupled collapse models, for a general noise power spectrum. For white noise with reduction rate $\lambda$, the phonon heating rate reduces to the standard formula, but for non-white noise with power spectrum $\lambda (\omega )$, the rate $\lambda$ is replaced by $\lambda_{\rm eff}=\frac{2}{3 \pi^{3/2}} \int d^3w e^{-\vec w^2} \vec w^2 \lambda(\omega_L(\vec w/r_c))$,with$\omega_L(\vec q)$ the longitudinal acoustic phonon frequency as a function of wave number $\vec{q}$, and with $r_C$ the noise correlation length. Hence if the noise power spectrum is cut off below ${\omega }_L(\mathrm{\mid }\vec{q}\mathrm{\mid }\sim r_c^{-1})$, the heating rate is sharply reduced.