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.

中文翻译：

---

被动和主动膜管的动力学。

利用Onsager的变分公式，我们得出了在小变形范围内松弛液膜管的动力学方程式，从而可以对比整个膜的溶剂粘度和由于膜不可压缩性引起的表面张力变化。我们计算弛豫率，在纯轴对称扰动的情况下恢复已知结果，并对高阶（方位角）m做出新的预测模式。我们通过使用各种渐近论证来分析这些模式的长波长和短波长限制。我们将随机项合并到适合描述被动热力和非平衡主动力的动力学方程中。我们导出了波动幅度，与主动波动相关的有效温度以及热和主动波动的功率谱密度的表达式。我们讨论了一种实验测定法，该测定法可以测量这些波动来推断出有源噪声的特性。最后，我们在活性膜的背景下更广泛地讨论我们的结果，并概述了该领域中一些未解决的问题。