Generalized Langevin dynamics for single beads in linear elastic network

TL;DR

This paper derives and compares different generalized Langevin equations for beads in elastic networks, confirming their relations and applying them to models like Rouse and ring polymers, including hydrodynamic interactions.

Contribution

It introduces new derivations of GLEs without normal modes, linking resistance and mobility kernels, and applies the theory to specific polymer models.

Findings

Derived GLEs for beads in elastic networks with resistance and mobility kernels.

Confirmed fluctuation-dissipation relations for both GLE representations.

Applied the theory to Rouse model, ring polymer, and elastic networks with hydrodynamics.

Abstract

We derive generalized Langevin equations (GLEs) for single beads in linear elastic networks. In particular, the derivations of the GLEs are conducted without employing normal modes, resulting in two distinct representations in terms of resistance and mobility kernels. The fluctuation-dissipation relations are also confirmed for both GLEs. Subsequently, we demonstrate that these two representations are interconnected via Laplace transforms. Furthermore, another GLE is derived by utilizing a projection operator method, and it is shown that the equation obtained through the projection scheme is consistent with the GLE with the resistance kernel. As simple examples, the general theory is applied to the Rouse model and the ring polymer, where the GLEs with the resistance and mobility kernels are explicitly derived for arbitrary positions of the tagged bead in these models. Finally, the GLE with the mobility kernel is also derived for the elastic network with hydrodynamic interactions under the pre-averaging approximation.