  Utilising Onsager's variational formulation, we derive dynamical equations
for the relaxation of a fluid membrane tube in the limit of small deformation,
allowing for a contrast of solvent viscosity across the membrane and variations
in surface tension due to membrane incompressibility. We compute the relaxation
rates, recovering known results in the case of purely axis-symmetric
perturbations and making new predictions for higher order (azimuthal)
$m$-modes. We analyse the long and short wavelength limits of these modes by
making use of various asymptotic arguments. We incorporate stochastic terms to
our dynamical equations suitable to describe both passive thermal forces and
non-equilibrium active forces. We derive expressions for the fluctuation
amplitudes, an effective temperature associated with active fluctuations, and
the power spectral density for both the thermal and active fluctuations. We
discuss an experimental assay that might enable measurement of these
fluctuations to infer the properties of the active noise. Finally we discuss
our results in the context of active membranes more generally and give an
overview of some open questions in the field.
