Partition function of a 3D vibrating string Is the partition function of a 3D vibrating string a sum of discrete energies, an integral of an energy continuum, or both?
$$
Z_{\text{disc}} = \sum_{k=1}^{\infty}g_ke^{-\beta E_k}
$$
or
$$
Z_{\text{cont}} = \int_{1}^{\infty}g_ke^{-\beta E_k}dk
$$
 A: Ideal string cannot be described in terms of canonical ensemble (with Boltzmann probability distribution).
The fact it is ideal means it has infinity of degrees of freedom and since each would have, according to the Boltzmann probability distribution, average energy 
$$\frac{1}{2}k_B T,$$
total energy of the string would be, on average, infinite. This is physically wrong result, so one should not apply the assumption of thermodynamic equilibrium to systems with infinite number of degrees of freedom.
However, if you replace the string by finite number of bodies connected with finite number of strings, this problem vanishes. The partition function of such system in classical statistical physics is
$$
Z = \int_\Gamma e^{-\frac{H(q,p)}{k_B T}}\,dqdp
$$
where $H$ is Hamiltonian function for the system and $\Gamma$ is the accessible phase space.
A: If you assume linearity and the string is finite, then the partition function is a sum of discrete energies: it is the same case as a string in 2D ( $y=f(x,t)$ ), but with two uncoupled dynamics: $(y,z)=(f(x,t),g(x,t))$. 
This can be shown by considering an oscillator in 2 dimensions with dynamics $\ddot{\vec{x}}=-k\vec{x}$: you will see you can separate it into two 1D oscillators (either $x,y$ modes or a radial and an axial mode).

More details on 2D oscillators here: http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node28.html
