Minimal Stochastic Model for Fermi's Acceleration

TL;DR

This paper presents a simple stochastic model that captures anomalous diffusion in Fermi's acceleration, providing explicit analytical solutions for velocity and position distributions, highlighting non-Gaussian behavior and scaling properties.

Contribution

It introduces a minimal stochastic model for Fermi's acceleration that is analytically solvable and exhibits anomalous diffusion with explicit probability distributions.

Findings

Derives explicit asymptotic PDFs for velocity and position.

Shows the diffusion is anomalous and non-Gaussian.

Identifies scaling laws with specific exponents.

Abstract

We introduce a simple stochastic system able to generate anomalous diffusion both for position and velocity. The model represents a viable description of the Fermi's acceleration mechanism and it is amenable to analytical treatment through a linear Boltzmann equation. The asymptotic probability distribution functions (PDF) for velocity and position are explicitly derived. The diffusion process is highly non-Gaussian and the time growth of moments is characterized by only two exponents $\nu_x$ and $\nu_v$. The diffusion process is anomalous (non Gaussian) but with a defined scaling properties i.e. $P(|{\bf x}|,t) = 1/t^{\nu_x}F_x(|{\bf x}|/t^{\nu_x})$ and similarly for velocity.