The peculiar response of Kelvin-Voigt chains with a free end

Abstract

We exactly solve a model of a heterogeneous chain of overdamped, harmonically coupled particles with momentum-conserving dissipation. Despite being governed by a non-symmetric drift operator, the system admits an analytical diagonalization by use of a forward-difference transformation. In case of one free end, the response matrix shows a peculiar staircase form: the response of particle $i$ to a force acting on particle $j$ is independent of the properties and the length of the chain-part between $i$ and $j$. For rank-deficient interaction matrices, the state space is decomposed into free and constrained subspaces. We demonstrate that this separation has clear physical consequences: the free subspace governs steady state responses, while the constraint subspace governs the relaxation after cessation of forcing. We further show that a Maxwell chain arises as a singular limiting topology of the Kelvin-Voigt chain. These results establish a framework for analyzing heterogeneous overdamped dynamics with momentum conservation.

14 pages, 8 figures