The generalized Langevin equation in harmonic potentials: Anomalous diffusion and equipartition of energy

TL;DR

This paper investigates the generalized Langevin equation with harmonic potentials and power law memory decay, revealing conditions for anomalous diffusion behaviors and energy equipartition, extending prior specific kernel results.

Contribution

It generalizes the understanding of energy equipartition and diffusion in GLEs with broad classes of memory kernels, including completely monotonic functions.

Findings

Displacement can be diffusive or superdiffusive in harmonic traps.

Energy equipartition holds under broad kernel conditions.

Results extend known cases like the Rouse kernel and power law kernels.

Abstract

We consider the generalized Langevin equation (GLE) in a harmonic potential with power law decay memory. We study the anomalous diffusion of the particle's displacement and velocity. By comparison with the free particle situation in which the velocity was previously shown to be either diffusive or subdiffusive, we find that, when trapped in a harmonic potential, the particle's displacement may either be diffusive or superdiffusive. Under slightly stronger assumptions on the memory kernel, namely, for kernels related to the broad class of completely monotonic functions, we show that both the free particle and the harmonically bounded GLE satisfy the equipartition of energy condition. This generalizes previously known results for the GLE under particular kernel instances such as the generalized Rouse kernel or (exactly) a power law function.