localization (and thus are not modes in need of stabilization via $\VExc$). We remark that the approximate interaction $\ExclVolu\,\delta(\PartPosi)$ is not, in practice, singular, and is instead regularized via a high wave vector cut-off.

At our coarse-grained level of description, the particles of the RLPM can be identified with polymers or small molecules, and the soft links can be identified with molecular chains that bind the molecules to one another. The potential for the soft links can be modeled as Gaussian chains $$\begin{aligned}
\label{EQ:VGC}
    \VLink^{(\textrm{GC})}(\vert r \vert) = \frac{\BoltCons T \vert r \vert^2}{2\LinkScal^2} ,\end{aligned}$$ i.e., a harmonic attraction, or a zero rest-length spring, of lengthscale $\LinkScal$ between the two particles. In making this coarse-graining one is assuming that microscopic details (e.g., the precise locations of the cross-links on the polymers, the internal conformational degrees of freedom of the polymers, and the effects of entanglement) do not play significant roles for the long-wavelength physics. In part, these assumptions are justified by studying more detailed models, in which the conformational degrees of freedom of the polymers are retained [@Goldbart1996]. However, we should point out that the precise form of $\VLink$ is not important for long-wavelength physics and, hence, for the elastic properties 