Brownian motion on a stochastic harmonic oscillator chain: limitations of the Langevin equation

Abstract

Diffusion of a Brownian particle along a linear chain of coupled stochastic harmonic oscillators is investigated using molecular dynamics (MD) and stochastic modeling. The latter technique is based on the Langevin equation (LE) derived by applying linear response theory to the chain degrees of freedom. When the coupling strength between the particle and the chain oscillators is comparable to or exceeds the chain coupling strength, the LE becomes inaccurate in its predictions of the diffusion coefficient value; however, it does reproduce qualitatively correctly the non-monotonic dependence of the diffusion coefficient on the particle-chain coupling strength, also found in MD simulations. The diffusion coefficient versus temperature curves determined from MD and Langevin simulations agree very well with each other at low temperatures. At high temperatures, the diffusion coefficient obtained from Langevin simulations is proportional to temperature, as predicted by Einstein’s relation. In contrast, the diffusion coefficient from MD is a non-linear function of temperature and is significantly greater than in Langevin simulations. Next, it is shown that Langevin description breaks down when the coupling between the chain oscillators is too strong. Finally, when an external constant force is applied to the particle, the Langevin description becomes qualitatively wrong. Namely, MD shows that the particle quickly detaches itself from the chain and moves at a constant acceleration due to the external force, whereas Langevin simulations predict constant terminal velocity of the particle.