Self-Similar Currents and Their Properties Based on the General Theory of Fractal Elements

This paper is a first step toward providing answers to the question of whether fractal pattern formation gives rise to power-law (fractional) kinetics and how such kinetics relate to geometric properties such as fractal dimension. The study focuses on Lichtenberg figures produced by high-voltage discharges on wood, a heterogeneous dielectric medium with anisotropic conductivity and variable moisture content. During breakdown, the discharge propagates through branching streamers and carbonization fronts, exhibiting scale-free growth, long-tailed waiting times, and memory effects. The associated current signals are analyzed using the theory of fractal elements developed by Nigmatullin and Chen. This framework allows complex self-similar waveforms to be decomposed into elementary fractal modes characterized by power-law exponents and amplitudes. The results show that the electrical response is governed by fractional dynamics encoded in these modes. However, no direct one-to-one relationship is found between the fractal dimension of the discharge patterns and the kinetic power-law exponents. This decoupling is attributed to the influence of the heterogeneous medium and the percolation pathways through which the discharge propagates.