  The Cryogenic Underground Observatory for Rare Events (CUORE) is the most
sensitive experiment searching for neutrinoless double-beta decay
($0\nu\beta\beta$) in $^{130}\text{Te}$. CUORE uses a cryogenic array of 988
TeO$_2$ calorimeters operated at $\sim$10 mK with a total mass of 741 kg. To
further increase the sensitivity, the detector response must be well
understood. Here, we present a non-linear thermal model for the CUORE
experiment on a detector-by-detector basis. We have examined both equilibrium
and dynamic electro-thermal models of detectors by numerically fitting
non-linear differential equations to the detector data of a subset of CUORE
channels which are well characterized and representative of all channels. We
demonstrate that the hot-electron effect and electric-field dependence of
resistance in NTD-Ge thermistors alone are inadequate to describe our
detectors' energy dependent pulse shapes. We introduce an empirical
second-order correction factor in the exponential temperature dependence of the
thermistor, which produces excellent agreement with energy-dependent pulse
shape data up to 6 MeV. We also present a noise analysis using the fitted
thermal parameters and show that the intrinsic thermal noise is negligible
compared to the observed noise for our detectors.
