Diffusional screening in treelike spaces: an exactly solvable diffusion-reaction model.

A renormalization approach is used to derive an analytic formula for the total current crossing the reactive surface of a Cayley tree of cylindrical tubes under a Helmholtz-type approximation to the full diffusion-reaction problem. We provide analytic conditions for the emergence of a plateau in the current-a region of maximum insensitivity of the current to variations in either the reaction rate (membrane permeability) or the diffusivity. The occurrence of such a plateau is associated with a partial screening regime wherein most of the active surface is screened to incoming diffusing particles. Large trees trade efficiency for fault tolerance, a valuable feature which may provide robustness to mammalian respiratory systems and tolerance to catalytic poisoning in chemical reactors.

Reverse engineering of oxygen transport in the lung: adaptation to changing demands and resources through space-filling networks.

The space-filling fractal network in the human lung creates a remarkable distribution system for gas exchange. Landmark studies have illuminated how the fractal network guarantees minimum energy dissipation, slows air down with minimum hardware, maximizes the gas- exchange surface area, and creates respiratory flexibility between rest and exercise. In this paper, we investigate how the fractal architecture affects oxygen transport and exchange under varying physiological conditions, with respect to performance metrics not previously studied.We present a renormalization treatment of the diffusion-reaction equation which describes how oxygen concentrations drop in the airways as oxygen crosses the alveolar membrane system. The treatment predicts oxygen currents across the lung at different levels of exercise which agree with measured values within a few percent. The results exhibit wide-ranging adaptation to changing process parameters, including maximum oxygen uptake rate at minimum alveolar membrane permeability, the ability to rapidly switch from a low oxygen uptake rate at rest to high rates at exercise, and the ability to maintain a constant oxygen uptake rate in the event of a change in permeability or surface area. We show that alternative, less than space-filling architectures perform sub-optimally and that optimal performance of the space-filling architecture results from a competition between underexploration and overexploration of the surface by oxygen molecules.

Heterogeneity explains features of "anomalous" thermodynamics and statistics.

Phenomena characterized by power-law probability distributions abound in nature and the applied sciences. We show that many of these power laws are well described by the Student, or t, distribution, and we discuss the origin of this universality based on three examples (Brownian motion, Knudsen diffusion in rough pores, and bubbly multiphase flow). These case studies are representative for a large class of systems with heterogeneous features, examples of which can be found from Earth sciences to astrophysics, and even in the social sciences. We show that common forms of polydispersity, such as polydispersity arising naturally as a result of aggregation-fragmentation phenomena, typically lie at the basis of the observed scaling. We conclude that complicated arguments based on long-range correlations or nonergodicity are often incorrect or misleading in explaining many naturally observed power laws and, in particular, those described by the Student distribution.