Finite-energy L\'evy-type motion through heterogeneous ensemble of Brownian particles

TL;DR

This paper introduces a Heterogeneous Ensemble of Brownian Particles (HEBP) model that captures anomalous diffusion with finite energy and moments, bridging the gap between Lévý flights and walks, and classical Brownian motion.

Contribution

The paper presents a novel HEBP model based on a linear Langevin equation that exhibits Lévý-like anomalous diffusion while maintaining finite energy and moments, unlike traditional Lévý processes.

Findings

HEBP displays power-law decaying PDF and long-range correlations.

It transitions from anomalous to normal diffusion over time.

HEBP maintains finite energy and satisfies the fluctuation-dissipation theorem.

Abstract

Complex systems display anomalous diffusion, whose signature is a space/time scaling $x\sim t^\delta$ with $\delta \ne 1/2$ in the Probability Density Function (PDF). Anomalous diffusion can emerge jointly with both Gaussian, e.g., fractional Brownian motion, and power-law decaying distributions, e.g., L\'evy Flights (LFs) or L\'evy Walks (LWs). LFs get anomalous scaling, but also infinite position variance and also infinite energy and discontinuous velocity. LWs are based on random trapping events, resemble a L\'evy-type power-law distribution that is truncated in the large displacement range and have finite moments, finite energy and discontinuous velocity. However, both LFs and LWs cannot describe friction-diffusion processes. We propose and discuss a model describing a Heterogeneous Ensemble of Brownian Particles (HEBP) based on a linear Langevin equation. We show that, for proper distributions of relaxation time and velocity diffusivity, the HEBP displays features similar to LWs, in particular power-law decaying PDF, long-range correlations and anomalous diffusion, at the same time keeping finite position moments and finite energy. The main differences between the HEBP model and two LWs are investigated, finding that, even if the PDFs are similar, they differ in three main aspects: (i) LWs are biscaling, while HEBP is monoscaling; (ii) a transition from anomalous ($\delta \ne 1/2$) to normal ($\delta = 1/2$) diffusion in the long-time regime; (iii) the power-law index of the position PDF and the space/time diffusion scaling are independent in the HEBP, while they both depend on the scaling of the inter-event time PDF in LWs. The HEBP model is derived from a friction-diffusion process, it has finite energy and it satisfies the fluctuation-dissipation theorem.