Scaling Limits of Interacting Particle Systems: The case of SOS and WASEP

Abstract

An interacting particle system is a stochastic process describing the evolution of finitely or infinitely many particles, each of which, in the absence of any interaction, would evolve like a Markov process. The particle system can be examined in different scales by grouping particles together and considering their average. Their behaviour is described by a Markov process on a microscopic scale, while on a macroscopic scale, it is represented by a partial differential equation. In the scaling limit, both the deterministic and stochastic properties are significant, and a stochastic differential equation characterizes the particle system.

The Weakly Asymmetric Simple Exclusion Process (WASEP) and the Solid On Solid (SOS) growth process are, though modelling different physical phenomena, related in that the WASEP can be seen as the discrete derivative of the SOS. It was thus sufficient to simulate only the WASEP and calculate the value of the SOS after the simulation finished. As expected, the simulations suggest systems with more sites have a smoother final distribution and take longer to converge. When the number of empty and occupied sites in the WASEP was close, the variance was higher, and convergence was faster.

In Bertini and Giacomins paper Stochastic Burgers and KPZ equations from Particle Systems, the Kardar-Parisi-Zhang equation was proven to be the hydrodynamic scaling limit to the SOS. Instead of considering the non-linear stochastic PDE directly, they used the Cole-Hopf transform to get the stochastic heat equation, for which the existence and uniqueness of a solution are known. A similar transform, the Gärtner transform, was used on a scaled and linearly interpolated SOS process. The transformed process was shown to converge to the stochastic heat equation, which implies that the SOS converges to the KPZ-equation in the hydrodynamic limit. The same method of proof was used to show the convergence of the WASEP to the stochastic Burgers equation.

The SOS belongs to the KPZ universality class. Models in this class are expected to be invariant under the scaling . Much of the research on KPZ universality has been done with the SOS since it has the necessary properties to be linearised with the Gärtner transform.

Popular Abstract

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.

• author: Heideman, Johanna
• supervisor: Filip Bar
• organization: Mathematics (Faculty of Engineering)
• course: FMAM05 20241
• year: 2024
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Interacting particle systems, Scaling limits, WASEP, SOS, KPZ-equation
• publication/series: Master's Theses in Mathematical Sciences
• report number: LUTFMA-3521-2023
• ISSN: 1404-6342
• other publication id: 2023:E77
• language: English
• id: 9162938
• date added to LUP: 2024-06-28 15:31:17
• date last changed: 2024-06-28 15:31:17