Diffusion is essential in biological processes, generating molecular fluxes that sustain life. Classically, these fluxes are described by Fick’s laws for random walks near thermal equilibrium. However, active systems locally inject energy into their surrounding medium, driving non-equilibrium diffusion for which a general theory remains elusive. We distinguish between two types of systems: active baths and active carpets. In the former, suspensions of active particles can enhance mixing and transport cargo using collective hydrodynamic entrainment. For active carpets, biological activity is highly concentrated on surfaces, such as biofilms. They can generate flows to attract nutrients from the bulk, by clustering and forming topological defects. Moreover, active carpets produce anisotropic and space-dependent diffusion, for which we derived generalized Fick’s laws. We solved two archetypal problems using these laws: First, considering sedimentation towards an active carpet, we find a self-cleaning effect where surface-driven fluctuations can repel particles. Second, considering diffusion from a source to an active sink, say nutrient capture by suspension feeders, we derived the enhanced molecular flux compared to thermal diffusion. Hence, our results could elucidate certain non-equilibrium properties of active coating materials and life at interfaces.

The fractional order “s” of a Lévy flight controls the algebraic decay of its corresponding jump length distribution. When the fractional order s is s>1/2 , s=1/2, or s<1/2 the corresponding one-dimensional Lévy flights is qualitatively similar to Brownian motion in one-, two-, and three-dimensions. In this talk I will describe how this correspondence emerges in the asymptotic analysis of two fractional problems: the characterization of spike equilibrium solutions to singularly perturbed reaction-diffusion systems, and the analysis of the first hitting-time (FHT) for a Lévy flight to a small target. These two fractional problems are inspired by questions in pattern formation and optimal foraging theory. Our asymptotic results provide qualitative insights into how the fractional order affects the linear stability of spike solutions, as well as how target sparsity determines the optimal fractional order minimizing the first hitting-time.