From the Underdamped Generalized Elastic Model to the Single Particle Langevin Description

Alessandro Taloni

doi:10.3390/math5010003

Abstract

The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages of the stochastic dynamics. We derive the generalized Langevin equation for a probe particle and we show that this equation reduces to the usual Langevin equation for Brownian motion, and to the fractional Langevin equation on the long-time limit.

Highlights

Linear stochastic systems describe a wealth of physical phenomena, ranging from polymer science to material science, where the elastic interactions inherent to the systems components are counterplayed by a friction dissipative force
Let us start by considering the GEM, including the inertial effects, i.e., the underdamped generalized elastic model (UGEM):
As a matter of fact, once plugged into the damping kernel expression (14) and into the noise propagator (15), they assure that the probe stochastic dynamics is described by a generalized Langevin equation such that

Summary

Introduction

Linear stochastic systems describe a wealth of physical phenomena, ranging from polymer science to material science, where the elastic interactions inherent to the systems components are counterplayed by a friction dissipative force (damping). The probe can represent a spot on a membrane surface, or a tracer particle among those composing a single file system, or a tagged monomer in polymeric chains This equation has the same structure as of an overdamped. In other words we plan to derive the equivalent of the FLE (5) for systems where inertial effects cannot be discarded This equation should properly reproduce any stochastic regime attained by the tagged particle, reducing to the FLE for long times

Underdamped Generalized Elastic Model

Generalized FD Relationship

Limiting Behaviours

Conclusions