  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.
