The common mathematical structure of noise, oxygen diffusion and LEDs

Marcel Filoche. What is the common point between oxygen diffusing in the lung to reach the blood through the alveolar barrier, a noise abatement wall aiming at damping the acoustic energy contained in the surrounding noise, or the behavior of electrons in modern light emitting diodes (LED), which are probably one of our low energy sources of light in the future ? Not much, apparently. They operate at completely different scales (tens of microns, meters, and nanometers, respectively), and their physical natures are miles apart. But they have in common an underlying mathematical structure. This structure is studied by an interdisciplinary field that includes harmonic analysis, which is concerned with the representation of functions as the superposition of basic waves (harmonics) ; elliptic PDEs, a specific type of partial differential equations ; and a geometric structure called Laplacian field. This field is of huge relevance in physics as half of the equations found in physics are wave equations (Maxwell equation, Schrödinger equation, Dirac equation, acoustic waves) or involve elliptic operators (diffusion equations, Poisson equation).

The consequence of this shared mathematical background is that analogous phenomena occur in all these systems : as we have shown with Pr. B. Sapoval from Ecole Polytechnique (France), oxygen molecules tend to concentrate and transfer to the blood in some specific regions of the alveolar interface, while deserting others (mathematicians would say that oxygen concentration does not “measure” the interface uniformly). This feature plays a key role in understanding the robustness of the lung. Acoustic waves do not spread uniformly but focus and remain for a long time in anfractuosities of a so-called noise abatement wall, where they are finally absorbed. There lies the secret of the Fractal® wall efficiency, designed from mathematical ideas. And electronic waves, as we have shown with Pr. S. Mayboroda from the University of Minnesota (USA), flow through very specific channels : light is not emitted uniformly in the structure. Understanding this is the key to unlock the performance of our future energy-saving lighting.

Let’s try to understand the common mathematical models that describe these phenomena. In the first case, diffusion means that oxygen molecules leave the air rich in oxygen and aim for the venous blood poor in oxygen. But in order to do so, these molecules have to cross the alveolar barrier, an enormous (100 m2) and highly intricate folded surface which separates air from blood and which is needed to breathe. In the second case, acoustic waves are pressure wave which propagate into the air. Sometimes, this sound is unwanted and we call it noise. To dampen the noise, we put on its way obstacles made out of absorbing material. The thing is that a geometrically complicated wall, full of bumps and anfractuosities, works incredibly more efficiently in absorbing the acoustic energy than a flat surface made of the same material.

Finally, in the third case, electrons move through the atomistic structure of matter. Layers of different semiconductor materials help confining them at specific locations where they are captured by atoms, giving away in the process their excess energy which is emitted as light. However, at the atomistic scale, electrons are not point-like particles or tiny hard spheres : they are quantum waves that spread, focus, diffract, or interfere with themselves. And the medium they travel through does not look to them as a perfectly ordered crystal. The materials used in LEDs are alloys whose local composition can strongly vary from one place to another. Interfaces between different materials are not flat either, and display roughness, disorder, and geometrical complexity. In short, electrons are waves trying to make their way through a very bumpy landscape (we call it a potential).

In mathematical terms, the equations that govern the evolution of all these systems contain the same differential operator. A differential operator is the generalization of the derivative of a function, but in spaces with several dimensions. It gives an assessment not only of the value of the function, but also of the way the function varies around each point. Also, these equations are linear. You can add to an existing population of oxygen molecules new oxygen molecules and the full population will still diffuse according to the same equation. You can superimpose acoustic waves (or two different sounds) and the sum will still form an acoustic wave (a new sound). You can even superimpose quantum waves : this is a key feature of quantum physics. Third, all processes occur in a very complex environment : convoluted interfaces between two different media or materials, disorder, fluctuations. And this geometrical complexity changes everything.

Amazingly, the study of the intrinsic mathematical properties of similar equations geometry gave us simultaneously invaluable insights into all these structures. These properties are universal, so, by studying mathematically the deep and intrinsic properties of elliptic equations, we could for instance increase the absorption of noise abatement structures by optimizing the shape of the wall, or control the propagation of waves, focusing them somewhere or on the contrary expelling them from specific locations.

Marcel Filoche is a researcher at the Laboratoire de Physique de la Matière Condensée (École Polytechnique)