Exact Generalized Langevin Dynamics of Pair Coordinates in Elastic Networks

TL;DR

This paper derives an exact homogeneous generalized Langevin equation for the relative coordinate of two beads in elastic networks, enabling systematic reduction of complex network dynamics to pair coordinates.

Contribution

It provides an analytical derivation of an exact hGLE for pair coordinates in elastic networks, expressing memory kernel and restoring force explicitly in terms of network matrices.

Findings

Explicit expressions for memory kernel and restoring force in terms of network matrices.

Derived a hGLE for inter-bead distance under small-displacement approximation.

Potential applications in modeling proteins and soft-matter systems.

Abstract

Generalized Langevin equations (GLEs) provide a powerful framework for describing slow dynamics in soft-matter systems, but deriving an exact homogeneous GLE (hGLE) for a reaction coordinate from an underlying many-body system remains generally difficult. Here, we analytically derive an exact hGLE for the relative coordinate of two tagged beads in arbitrary elastic networks. The memory kernel and effective restoring force are expressed explicitly in terms of the network matrices, thereby providing a systematic reduction of the high-dimensional network dynamics to a pair coordinate. Within the small-displacement approximation, we further derive a hGLE for the inter-bead distance, a central observable in distance-sensitive single-molecule experiments. These results therefore have broad potential applications in modeling proteins and other soft-matter systems.