Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles

Abstract

Complex systems are known to display anomalous diffusion, whose signature is a space/time scaling x ∼ t δ with δ = 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., Lévy Flights (LFs) or Lévy Walks (LWs). LFs get anomalous scaling, but also infinite position variance and, being jumps of any size allowed even at short times, also infinite energy and discontinuous velocity. LWs are based on random trapping events, resemble a Lévy-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 and do not take into account the role of strong heterogeneity in many complex systems, such as biological transport in the crowded cell environment. 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, longrange 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 (δ = 1/2) to normal (δ = 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.

Gaussian Processes in Complex Media: New Vistas on Anomalous Diffusion

“…In this paper we consider a special class of RSGPs called generalized gray Brownian motion (ggBm), that is defined by using the fractional Brownian motion as Gaussian process [70][71][72][73][74][75]. For other form of randomly-scaled Gaussian process we refer the reader to Sliusarenko et al [59]. Hence, we consider the following class of processes:…”

“…formula (59) reduces to the integral formula (A.10) for the symmetric space-time fractional diffusion equation. In terms of random variables it follows that [56]…”

“…Other authors follow a somewhat different approach based on random-scaled Gaussian processes (RSGPs) [38,[56][57][58][59], which are physically based on a recently proposed model where interparticle heterogeneity is explicity described through a population of scales characterizing the dynamical parameters of particle diffusive motion. This modeling approach has been denoted as heterogeneous ensemble of Brownian particles (HEBP) and has been developed on the basis of a Langevin model [57][58][59]. The HEBP model is then based on the Gaussian-Wiener process and, thus, on trajectories that are strongly continuous in the stochastic sense [60], while anomalous diffusion emerge as a consequence of heterogeneity.…”

First passage statistics for diffusing diffusivity

“…where D is the random and constant diffusion coefficient and W(t) is the d-dimensional Wiener process or standard Brownian motion. Note that while the DD model represents the heterogeneity of the medium in some mean field sense [22] the ggBM model describes an heterogeneous ensemble of particles [29] .…”

The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach

“…The heterogeneous ensemble of Brownian particle (HEBP) approach [1,2] is based on the idea that a population of scales in the system in which particles are diffusing may generate the anomalous diffusing behaviour observed in many physical and biological systems [3][4][5][6]. Long time and space correlation, characteristics of many anomalous diffusion processes [7][8][9], are often described through the introduction of memory kernels and integral operators [10,11], as the fractional derivatives are [12], leading in general to non-Markovianity and/or non-locality of the processes.…”