  In this work we first briefly review some of the mutual friction effects on
vortex lines and rings that were obtained in the context of the truncated
Gross-Pitaevskii equation in references Krstulovic \& Brachet [Phys.~Rev.~E
\textbf{83}(6), 066311 and Phys.~Rev.~B \textbf{83}132506 (2011)], with
particular attention to the anomalous slowdown of rings produced by thermally
excited Kelvin waves. We then study the effect of mutual friction on the
relaxation and fluctuations of Kelvin waves on straight vortex lines by
comparing the results of full $3D$ direct simulations of the truncated
Gross-Pitaevskii equation with a simple stochastic
Local-Induction-Approximation model with mutual friction and thermal noise
included. This new model allows us to determine the mutual friction coefficient
$\alpha$ and $\alpha'$ for the truncated Gross-Pitaevskii equation.
