Multifractal Collision Spectrum of Ballistic Particles with Fractal Surfaces

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.