Finite-inertia effects in Langevin dynamics of a lopsided elastic dumbbell using exponential-time differencing schemes

Inertia effects in the Langevin dynamics of a lopsided elastic dumbbell are investigated using exponential-time-differencing (ETD) integrators for the corresponding stiff stochastic equations in the small-mass limit. Starting from the bead-level underdamped Langevin model, we formulate the dynamics in modal coordinates, highlighting two distinct friction scales: an additive friction ζ trans = ζ 1 + ζ 2 controlling translation ( ζ i , i = 1, 2 are the friction coefficient of bead i ), and an effective internal friction 1/ ζ eff = 1/ ζ 1 + 1/ ζ 2 controlling configurational relaxation, with relaxation time τ R = ζ eff / H for a Hookean spring of stiffness H . We benchmark ETD against Euler–Maruyama and overdamped Brownian dynamics using equilibrium statistics, time-domain autocorrelations, and frequency-domain power spectra of the end-to-end vector. When time is rescaled by τ R , configurational and orientational relaxation curves collapse across asymmetry ratios, showing that the dominant long-time structural dynamics remains close to the overdamped description. Inertial signatures are instead confined to short-time transients, high-frequency modifications of the configurational spectrum, and a transient coupling between translational and internal modes. This study provides a practical and accurate route for lopsided dumbbells across overdamped and weakly underdamped regimes, and clarifies how mass and friction asymmetry affect the translational and internal dynamics.