Multifractal Collision Spectrum of Ballistic Particles with Fractal Surfaces

TL;DR

This study investigates the multifractal properties of collision distributions of ballistic particles on irregular fractal surfaces, revealing a trinomial multifractal structure unaffected by micro-roughness.

Contribution

It introduces a numerical analysis of collision spectra showing a trinomial multifractal measure for particles interacting with fractal surfaces, independent of surface micro-roughness.

Findings

Collision measure is multifractal even with surface micro-roughness.

The multifractal spectrum consists of two parabolas, indicating a trinomial structure.

Surface micro-roughness affects the information dimension but not the fundamental trinomial nature.

Abstract

Ballistic particles interacting with irregular surfaces are representative of many physical problems in the Knudsen diffusion regime. In this paper, the collisions of ballistic particles interacting with an irregular surface modeled by a quadratic Koch curve, are studied numerically. The $q$ moments of the source spatial distribution of collision numbers $\mu(x)$ are characterized by a sequence of ``collision exponent'' $\tau(q)$. The measure $\mu(x)$ is found to be multifractal even when a random micro-roughness (or random re-emission) of the surface exists. The dimensions $f(\alpha)$, obtained by a Legendre transformation from $\tau(q)$, consist of two parabolas corresponding to a trinomial multifractal. This is demonstrated for a particular case by obtaining an exact $f(\alpha)$ for a multiplicative trinomial mass distribution. The trinomial nature of the multifractality is related to the type of surface macro-irregularity considered here and is independent of the micro-roughness of the surface which however influence the values of $\alpha_{min}$ and $\alpha_{max}$. The information dimension $D_I$ increases significantly with the micro-roughness of the surface. Interestingly, in contrast with this point of view, the surface seems to work uniformly. This correspond to an absence of screening effects in Knudsen diffusion.