Multifractal analysis of heat transport in DLA structures

Multifractal analysis originated from the desire to obtain a deeper understanding of materials that follow random fractal structures. Complex systems display diverse degrees of irregularity and self-similarity on multiple scales that cannot be adequately described by a single fractal dimension due to uncertainties as well as deterministic changes in structure at different length and time scales. The multifractal spectrum measures the distribution of a specific property across different scales or levels within a system using a range of exponents. These properties can encompass anything from turbulence intensity in fluid flow to the distribution of heat flow through a solid. Each exponent in the spectrum corresponds to a particular level of irregularity, and a broad range of exponents indicates more complexity. Multifractal measures have a range of applications, from biology to mechanical engineering, among many others. In multifunctional materials used in adaptive systems, a more accurate description of material properties, such as viscoelasticity, heat transport, phase transformations, often involves considering information about their underlying fractal material structure. This study goes beyond fractals to include randomness in terms of multifractal measures. This involves analysis of the Renyi entropy. We evaluate this approach by examining heat transport in DLA structures which display multifractal properties. Renyi entropy is relevant here due to its entropy order parameter's close connection to the multifractal spectrum. In fact, the multifractal spectrum can be directly calculated through the generalized Renyi entropy dimension and a box-counting process. We investigate the multifractality of different DLA structures and leverage this insight to better understand heat diffusion processes using fractal-order diffusion equations; however, the methodology is more broadly applicable to a wide range of multiphysics problems involving energy transport over complex media.