Correlations in a Generalized Elastic Model: Fractional Langevin Equation Approach

TL;DR

This paper demonstrates that the Generalized Elastic Model (GEM) and the Fractional Langevin Equation (FLE) frameworks are equivalent in describing the correlated stochastic dynamics of many-body systems, using Fox H-functions for correlation analysis.

Contribution

It establishes the equivalence between GEM and FLE descriptions for systems with spatial interactions, and introduces Fox H-functions as a useful tool for analyzing correlations.

Findings

Correlation functions from GEM and FLE match.

Fox H-function formalism effectively describes correlation properties.

FLE approach provides a consistent stochastic description.

Abstract

The Generalized Elastic Model (GEM) provides the evolution equation which governs the stochastic motion of several many-body systems in nature, such as polymers, membranes, growing interfaces. On the other hand a probe (\emph{tracer}) particle in these systems performs a fractional Brownian motion due to the spatial interactions with the other system's components. The tracer's anomalous dynamics can be described by a Fractional Langevin Equation (FLE) with a space-time correlated noise. We demonstrate that the description given in terms of GEM coincides with that furnished by the relative FLE, by showing that the correlation functions of the stochastic field obtained within the FLE framework agree to the corresponding quantities calculated from the GEM. Furthermore we show that the Fox $H$-function formalism appears to be very convenient to describe the correlation properties within the FLE approach.