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

Abstract

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 at 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 ${\zeta }_{\mathrm{trans}}={\zeta }_1+{\zeta }_2$ controlling translation (${\zeta }_i,i=1,2$ are the friction factor on bead $i$), and an effective internal friction $1/{\zeta }_{\mathrm{eff}}=1/{\zeta }_1+1/{\zeta }_2$ controlling configurational relaxation, with relaxation time ${\tau }_R={\zeta }_{\mathrm{eff}}/H$ 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 ${\tau }_R$, 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 clarify how mass and friction asymmetry affect the translational and internal dynamics.

References in corpus (2)

• High-Order Schemes of Exponential Time Differencing for Stiff Systems with Nondiagonal Linear Part
• Lopsided Rigid Dumbbell Rheology from Langevin Equation: A Graduate Tutorial