The Fractional Langevin Equation: Brownian Motion Revisited

TL;DR

This paper revisits Brownian motion using the fractional Langevin equation, incorporating hydrodynamic effects and fractional calculus to provide more realistic models of particle dynamics and diffusion behavior.

Contribution

It introduces a fractional Langevin equation framework that accounts for hydrodynamic backflow effects, offering analytical expressions for correlation functions and displacement, enhancing the realism of Brownian motion modeling.

Findings

Velocity correlation decays as t^{-3/2}, slower than exponential

Mean squared displacement shows normal diffusion at long times

Retarding effects may mimic anomalous diffusion in experiments

Abstract

We have revisited the Brownian motion on the basis of the fractional Langevin equation which turns out to be a particular case of the generalized Langevin equation introduced by Kubo on 1966. The importance of our approach is to model the Brownian motion more realistically than the usual one based on the classical Langevin equation, in that it takes into account also the retarding effects due to hydrodynamic backflow, i.e. the added mass and the Basset memory drag. On the basis of the two fluctuation-dissipation theorems and of the techniques of the Fractional Calculus we have provided the analytical expressions of the correlation functions (both for the random force and the particle velocity) and of the mean squared particle displacement. The random force has been shown to be represented by a superposition of the usual white noise with a "fractional" noise. The velocity correlation function is no longer expressed by a simple exponential but exhibits a slower decay, proportional to $t^{-3/2}$ as $t \to \infty$, which indeed is more realistic. Finally, the mean squared displacement has been shown to maintain, for sufficiently long times, the linear behaviour which is typical of normal diffusion, with the same diffusion coefficient of the classical case. However, the Basset history force induces a retarding effect in the establishing of the linear behaviour, which in some cases could appear as a manifestation of anomalous diffusion to be correctly interpreted in experimental measurements.