Complex Fluids in a Multifractal Space: Scale Covariance and the Emergence of the Fractal Force

TL;DR

This paper introduces a new mathematical framework using multifractal geometry to better understand complex systems like turbulent fluids and biological organisms.

Contribution

The novel approach uses scale covariance and non-differentiable multifractal curves to model complex system dynamics.

Findings

The conservation of momentum is reformulated as a geodesic equation in multifractal space.

Complex velocity fields are decomposed into differentiable and non-differentiable components using a singularity spectrum f(α).

Non-differentiability is suggested to improve predictions in fields like oncology and geophysics.

Abstract

Complex systems—ranging from biological organisms to turbulent fluids—exhibit multiscale heterogeneity and intermittency that traditional, differentiable calculus fails to adequately capture. Therefore, we propose a mathematical framework for analyzing complex system dynamics by assimilating the trajectories of structural units to continuous but non-differentiable multifractal curves. Utilizing the scale covariance principle, the authors recast the conservation of momentum as a geodesic equation within a multifractal space. This approach naturally separates the complex velocity field into differentiable and non-differentiable scale resolutions, where the balance of multifractal acceleration, convection, and dissipation is parametrized by a singularity spectrum f(α). We also discuss broad interdisciplinary implications, because, in our opinion, non-differentiability can enhance predictive capabilities in various fields such as oncology, pharmacology, and geophysics.

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