Emergence via hydrodynamic limits

When a drop of ink falls into a glass of water, the ink spreads throughout the water. At the molecular level, the water and ink molecules behave randomly. At the macroscopic level (the level we can observe), however, the ink always spreads out. The exact trajectory of the ink is different every time, but we know that its spread follows a diffusion equation. This is an example of chaotic behavior at the microscopic scale giving rise to predictable behavior at the macroscopic scale. This is emergence.
This phenomenon is common, not only in ink in water. Examples are people fleeing a building, but also bees and birds flying in swarms.
In this presentation, we study the same phenomenon using a mathematical model in which particles move around randomly according to simple rules. By zooming out so far that individual particles can no longer be seen and only the density remains visible, it becomes clear that the density follows a deterministic pattern. This pattern is described by a partial differential equation and is known as the hydrodynamic limit.
Proving the existence of the hydrodynamic limit is challenging. One technique used to establish it is duality. This technique makes it possible to study a single particle instead of all the particles simultaneously. This simplifies the system enough to prove the hydrodynamic limit. In doing so, we demonstrate emergence.

Phine Schikhof is a sophomore PhD student at TU Delft. Her research focuses on particle systems and their hydrodynamic limits. In particular, she is interested in particle systems in a space that changes over time itself. You can remember that certain pieces of space become easier or more difficult to pass or even temporarily closed. But it may also be that the space itself moves, so there is more or less space for particles and the subsurface can drag the particles with it. The moving surface causes allowed movements to change over time in one place. Because of this kind of dynamic, the techniques to prove the hydrodynamic limit for a non-moving space are not always usable anymore. At the moment Phine is therefore mainly expanding the existing techniques so that they also work for particle systems in a space that moves over time.