Anomalous energy losses in fractal medium

TL;DR

This paper derives an equation for energy loss distribution of particles in fractal media, revealing anomalous drift behavior characterized by Mittag-Leffler renewal processes and power-law dependence on distance.

Contribution

It introduces a novel mathematical model describing energy losses in fractal media with heterogeneities, highlighting the role of fractal dimension in anomalous drift.

Findings

Energy losses follow Mittag-Leffler renewal process in fractal media.

Average energy loss exhibits anomalous drift with power-law dependence.

The drift exponent relates directly to the fractal dimension D.

Abstract

We derive equation describing distribution of energy losses of the particle propagating in fractal medium with quenched and dynamic heterogeneities. We show that in the case of the medium with fractal dimension $2<D<3$ the losses of energy are described by the Mittag-Leffler renewal process. The average energy loss of the particle experiences anomalous drift $\Delta\sim x^{\alpha}$ with power-law dependence on the distance $x$ from the surface and exponent $\alpha=D-2$.