Luke Wilson | 2024 I.S. Symposium

Name: Luke Wilson
Title: Anisotropic and Relativistic Diffusion with Partial Differential Equations
Major: Mathematics; Physics
Advisor: Michael Bush; Megan Nieberding

Partial differential equations are versatile mathematical tools that are used to model a wide variety of physical systems. Among these are diffusion systems, where a substance “spreads out” over time. The heat equation, one of the best-known partial differential equations, can be used to model the vast majority of diffusion problems. Some systems, however, carry an inherent anisotropy where different spatial directions must be treated differently. There are several adjusted diffusion models widely used to model anisotropic systems. Furthermore, it appears that the classical diffusion models, both isotropic and anisotropic, violate a key result of special relativity: that nothing can move faster than the speed of light. In this thesis, a simple deterministic equation is introduced to model a constant “drift” anisotropy. That equation is then merged with an existing relativistic equation to produce a model that combines anisotropic and relativistic diffusion. Solutions to this equation and others are obtained in various physical contexts. The combined model is shown to be compatible with relativity and its diffusion-like behavior is demonstrated. Throughout, particular attention is paid to the qualitative behavior of the solutions and the physical consequences of that behavior.

Posted in Symposium 2024 on April 22, 2024.