Generalized Langevin equations (GLEs) provide a powerful framework for describing slow dynamics in soft-matter systems, but deriving an exact GLE for a reaction coordinate from an underlying many-body system remains generally difficult. Here, we analytically derive an exact coupled GLE for the overdamped dynamics of two tagged beads in arbitrary harmonic elastic networks. We further derive an exact GLE for their relative coordinate. 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 GLE for the interbead 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.

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• Brownian motion
• Fluctuation-dissipation theorem
• Polymers
• Langevin equation
• Non-Markovian processes