state, in relation to our results of low-frequency vibrational spectra.

System Description
==================

Coarse-grained MD simulations are performed by using the Kremer–Grest model [@Kremer:1990iv], which treats polymer chains as linear series of monomer beads (particles) of mass $m$. Each polymer chain is composed of $L$ monomer beads, and two cases are considered in this study: long chain length with $L=50$ and short chain length with $L=3$. In a three-dimensional cubic simulation box under periodic boundary conditions, $N_\mathrm{p} = 5000$ and $4998$ is defined as the total number of monomers for $L=50$ and $L=3$ respectively, which means that the number of polymeric chains is $N_\mathrm{p}/L = 100$ for $L=50$ and $1666$ for $L=3$.

The polymer chain is modeled by three types of inter-particle potentials as follows. Firstly, all the monomer particles interact via the LJ potential: $$U_\mathrm{LJ}(r) = 4\varepsilon_\mathrm{LJ} \left[ \left( \frac{\sigma}{r}
                        \right)^{12} - \left(
                        \frac{\sigma}{r} \right)^{6}
                     \right],
\label{eq:LJ}$$ where $r$ is the distance between two monomers, $\sigma$ is the diameter of monomer, and $\varepsilon_\mathrm{LJ}$ is the energy scale of the LJ potential. The LJ potential is truncated at the cut-off distance of $r_c = 2.5
\sigma$, where the potential and the force (first derivative of the potential) are shifted to 