Johanna Heideman will be presenting her MSc thesis "Scaling Limits of Interacting Particle Systems: The Case of SOS and WASEP"

Title:  Scaling Limits of Interacting Particle Systems: The Case of SOS and WASEP

Summary: The Weakly Asymmetric Simple Exclusion Process (WASEP) is a model used to, for example, describe the internal movement of a collection of gas molecules with the assumption that space is divided into discrete spaces. These spaces have room for one molecule and can be either occupied or empty. Here, the WASEP is modelled in one dimension. After a random amount of time, a molecule will attempt to move one step to the left or one step to the right. If the neighbouring spot is empty, it will succeed. Regardless of whether it moved or not, it will again wait for a random time to move. Weakly Asymmetric means that the particles are slightly more likely to want to move left than right. No matter where the particles were initially, after sufficient time, the average concentration of particles at different places on the line will stabilize. How fast this happens depends on how many parts the space is divided into and the proportion of occupied and empty spaces. In a larger system, where space is divided into more spaces, convergence is slower. On the other hand, when there are approximately the same number of occupied and empty spaces, it will converge faster. 

The Solid On Solid (SOS) process is a growth process. Among other phenomena, it describes how a crystal forms on a surface. One at a time, particles deposit on, or evaporate from, a place on the border of the crystal and the surrounding material. As deposition occurs slightly more often than evaporation, the crystal will grow. The SOS enforces a single-step constraint on the growth. It means that no molecules at the border are at the same height as their immediate neighbours; they are either one step above or below. As a consequence, deposition is only possible at places where both neighbours are one step higher, i.e. a local minimum. Similarly, the only particles that can evaporate are the ones that are higher than their surroundings. 

These two models are related. A molecule moves one step to the left in the WASEP at the same rate as a particle is deposited in the SOS. Further, movement to the left in the WASEP is possible when there is a particle in the original space and none in the left neighbouring space. When this occurs, there is a local minimum in the SOS. 

The WASEP and SOS models describe physical phenomena on a microscopic scale, where individual molecules are detectable. When zooming out, only the average motion is visible. Instead of measuring individual molecules, temperature, volume, and pressure describe the gas. The crystal grows continuously, and the border appears smooth. There is mathematical proof that when appropriately scaling space and time and increasing the number of spaces in the modelled system with time, the particle system models will converge to stochastic differential equations.

Supervisors: Carina Geldhauser, Filip Bar

Examiner: Sara Maad Sasane