The hydromechanic approach to complex diffusion

A tiny particle, just a few micrometers in size, when suspended in a liquid at rest, never remains still. Instead, it moves in an irregular and unceasing way: this is what is known as Brownian motion. It is characterized by a linear scaling of the mean squared displacement (MSD) together with Gaussian displacement distributions. In complex fluids, however, systematic deviations from these signatures have been widely observed, involving either both behaviors simultaneously or each one individually. Understanding the physical origin of these anomalies, that we may term complex diffusion phenomena, opens new avenues for exploring the microscopic dynamics that govern transport and relaxation processes in heterogeneous media. Such an understanding is crucial not only for unveiling the fundamental mechanisms underlying molecular motion in biological systems, but also for developing a unified framework to describe transport in soft matter, disordered materials, and active or viscoelastic environments. Most models used to describe these deviations rely on generalizations of the classical random walk, effectively performing a coarse-graining of the fluid’s reaction on the particle. This influence manifests itself in the specific form of the deviation from the standard random walk behavior, yet it remains implicit in the model formulation and is not directly represented in its physical structure. Including explicitly the coupling between the fluid and the particle means adopting a hydromechanic approach, that is, the study of the interactions between fluids and solid bodies in order to derive the mechanics of the bodies resulting from the fluid reaction, and the hydrodynamics of the fluid resulting from the reaction of the bodies. In this thesis, focusing on the first of these two aspects, we present, for the first time, a systematic formulation of the hydromechanic approach to complex diffusion. This objective is developed across the first three chapters, each presenting a set of original results. In the second Chapter (following the introductive one), we introduce a three-signature analysis, encompassing the MSD, displacement distributions, and the velocity autocorrelation function (VACF), that for the first time maps qualitative experimental patterns to specific particle–fluid interaction mechanisms. It should be noted that, at present, VACFs can be reliably measured only in simple fluids; however, it is reasonable to expect that in the coming years this quantity will also become experimentally accessible in complex fluids. In the third Chapter, we propose a new fluctuation–dissipation relation, termed the Global Effective Fluctuation–Dissipation Relation, which links the long-term diffusion coefficient to the effective friction factor. This relation is designed to be readily testable in laboratory conditions and provides a discriminating tool among different interpretative models of particle motion. Finally, in the fourth Chapter, we employ the 12 hydromechanic approach to derive the first theoretical explanation of the phenomenon known as Fickian yet non-Gaussian diffusion (FnGD) in the regime where the particle perceives the fluid as complex yet homogeneous. This result is of fundamental importance, as it provides for the first time a mechanistic explanation of the phenomenon when the particle size is such that it does not directly probe the medium’s heterogeneity. The same chapter also includes a first concise review of the main experimental results on FnGD reported to date. The second part of the thesis turns to a model system for complex diffusion: hydrogels (cross- linked polymer networks that retain large amounts of water while exhibiting soft, viscoelastic, and poroelastic behavior). Specifically, the fifth chapter explores nanoparticle transport within physically cross-linked alginate hydrogels, used here as tissue-mimicking materials, through epi- fluorescence microscopy. We analyze particle motion across increasing cross-linking levels, tuned by raising salt concentration. We introduce a new MSD-based classifier that segments trajectories into trapping, linear, and anomalous regimes, enabling direct comparison between low- and high-salt conditions. With stronger cross-linking, we observe the survival of only the faster subpopulation and a higher fraction of trapped particles. Finally, in the sixth Chapter we propose the first method to infer accessible porosity directly from standard z-projections by recasting anomalous transport into an effective Brownian clock: each connected accessible cluster is assigned an anomalous exponent as a proxy for local porosity. We quantify information loss due to finite depth of field away from the quasi-2D regime and provide practical rules to trade volumetric coverage against axial occlusion. The framework delivers conservative upper/lower bounds on porosity that tighten with longer acquisitions or complementary channels. As a separate contribution, we present what appears to be the first derivation of the Hausdorff dimension for an anomalous stochastic process.