# 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.

## Key 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.