I am reading through a paper. A tubular membrane, submitted to tension $\sigma$ acting as a Lagrange multiplier to conserve area, fluctuates around a cylindrical shape of length L and radius R. Parametrisation is as follows, \begin{equation} \mathbf{r}(\phi,\zeta)= (x,y,z) = R([1+u(\phi,\zeta)]\cos{(\phi)}, [1+u(\phi,\zeta)]\sin{(\phi)}, \zeta) \end{equation} where $0 \leq \phi \leq 2\pi$, $0 \leq \zeta \leq L/R$ and $u(\phi,\zeta)$ is the deformation field.
Using the Canham-Helfrich Hamiltonian, $F= \int dS\left( \frac{1}{2} \kappa H^2 + \sigma \right)$ one can obtain expressions, to second order, for the excess energy.
Long story short, the excess energy can be represented in Fourier modes from which you obtain the mean squared amplitude using the equipartition of energy: \begin{equation} \langle \lvert u_{m,n} \rvert ^2 \rangle = \frac{k_B T}{\kappa} \frac{1}{(m^2 -1)^2 + \bar{q}^2 (\bar{q}^2 + 2m^2)} \end{equation}
The correlation function can also be found: \begin{align} C((\phi - \phi '),( \zeta - \zeta')) &= \langle u(\phi,\zeta)u(\phi ', \zeta ') \rangle \\ &= \frac{k_B T R}{2 \pi \kappa L} \sum_{m,\bar{q}} \frac{\cos(m(\phi - \phi '))cos(\bar{q}(\zeta - \zeta '))}{(m^2 -1)^2 + \bar{q}^2 (\bar{q}^2 + 2m^2)} \end{align}
So my questions are:
1) What is the meaning of the mean squared amplitude and the correlation function?
2) How can I obtain a fluctuation spectrum from them?
The paper can be found here: http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.98.018103
Many thanks in advance.