Fractal growth processes frequently emerge in systems characterized by anomalous transport, memory effects, and long-range correlations, features that exponential relaxations fail to capture. This motivates a central question: does fractal pattern formation give rise to power-law (fractional) kinetics, and if so, how is the fractional order linked to geometric properties such as fractal dimension? The authors explore this question through various experiments and the proposed theory of fractal elements.

Theisstudy focuses on Lichtenberg figures generated by high-voltage discharges on wood surfaces, a striking example of fractal pattern formation in a heterogeneous dielectric medium. When a wooden substrate with anisotropic conductivity and moisture variability is subjected to electric fields above its breakdown threshold, the discharge propagates through branching streamers and carbonization fronts. These fronts display long-tailed waiting-time distributions and memory-dependent evolution.

Experimental evidence reveals that the discharge does not advance with constant velocity or classical exponential behaviour. This raises a key question: Does the electrical current measured during breakdown also exhibit fractional behaviour?

To investigate this, the authors use the recent theoretical framework of Nigmatullin and Chen (2023), which provides a rigorous method for analyzing complex, self-similar signals. Their “general theory of fractal elements” shows that an experimental waveform can be decomposed into elementary self-similar modes, each associated with a power-law scaling exponent and amplitude. This decomposition reveals that the current signal is a structured combination of fractal components encoding the underlying fractional dynamics.

Essentially, the essence of this theory is decomposition of a complex self-similar signal on a combination of fractal modes governed by a set of power-law exponents that can be confirmed on many self-similar processes (in time and space). The relationship between fractal dimension and power-law exponents governed by fractal dynamics is not found due to the influence of the medium structure, where the discharge current is propagated.