One-dimensional acoustic phonon heat capacity
= One-dimensional acoustic phonon heat capacity
{title2=$C/\ell=\pi k_B^2T/(3\hbar v)$}
A single acoustic branch $\omega=v|k|$ has thermal energy per length $U_T/\ell=\int dk\,\hbar v|k|/[2\pi(e^{\beta\hbar v|k|}-1)]=\pi(k_BT)^2/(6\hbar v)$. Differentiating gives the displayed linear <heat capacity>. The regime requires <temperatures> below the microscopic <phonon> cutoff but above the finite-size level spacing, with the translational <zero mode> treated separately.