Surveying Diffusion in Complex Geometries. An Essay

TL;DR

This paper reviews the challenges and recent advances in understanding diffusion within complex three-dimensional geometries, emphasizing new computational and spectral methods to analyze diffusion in natural and industrial structures.

Contribution

It introduces a novel approach combining fast random walk algorithms with spectral tools to study diffusion in complex geometries, advancing the understanding of diffusion processes in irregular structures.

Findings

Developed a new combined computational-spectral method for diffusion analysis.

Applied methods to model geometries like von Koch boundaries and Kitaoka acinus.

Enhanced understanding of diffusion effects like screening and localization in complex domains.

Abstract

The surrounding world surprises us by the beauty and variety of complex shapes that emerge from nanometric to macroscopic scales. Natural or manufactured materials (sandstones, sedimentary rocks and cement), colloidal solutions (proteins and DNA), biological cells, tissues and organs (lungs, kidneys and placenta), they all present irregularly shaped "scenes" for a fundamental transport "performance", that is, diffusion. Here, the geometrical complexity, entangled with the stochastic character of diffusive motion, results in numerous fascinating and sometimes unexpected effects like diffusion screening or localization. These effects control many diffusion-mediated processes that play an important role in heterogeneous catalysis, biochemical mechanisms, electrochemistry, growth phenomena, oil recovery, or building industry. In spite of a long and rich history of academic and industrial research in this field, it is striking to see how little we know about diffusion in complex geometries, especially the one which occurs in three dimensions. We present our recent results on restricted diffusion. We look into the role of geometrical complexity at different levels, from boundary microroughness to hierarchical structure and connectivity of the whole diffusion-confining domain. We develop a new approach which consists in combining fast random walk algorithms with spectral tools. The main focus is on studying diffusion in model complex geometries (von Koch boundaries, Kitaoka acinus, etc.), as well as on developing and testing spectral methods. We aim at extending this knowledge and at applying the accomplished arsenal of theoretical and numerical tools to structures found in nature and industry.