Foundation of Fractional Langevin Equation: Harmonization of a Many Body Problem

TL;DR

This paper derives a fractional Langevin equation from first principles for a tracer particle in a many-body system, providing a fundamental basis for models of anomalous dynamics.

Contribution

It introduces a harmonization technique to connect microscopic interactions with fractional Langevin equations, clarifying their foundational assumptions.

Findings

Derivation of a fractional Langevin equation from microscopic dynamics

Establishment of a link between many-body interactions and anomalous diffusion

Insight into the fundamental assumptions of phenomenological models

Abstract

In this study we derive a single-particle equation of motion, from first-principles, starting out with a microscopic description of a tracer particle in a one-dimensional many-particle system with a general two-body interaction potential. Using a new harmonization technique, we show that the resulting dynamical equation belongs to the class of fractional Langevin equations, a stochastic framework which has been proposed in a large body of works as a means of describing anomalous dynamics. Our work sheds light on the fundamental assumptions of these phenomenological models.