Hydrodynamic Limits for Long Range Asymmetric Processes and Probabilistic Opinion Dynamics
Publisher
The University of Arizona.Rights
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.Abstract
This dissertation focuses on two problems that can be modeled by interacting particle systems: hydrodynamics and opinion dynamics. From a hydrodynamic standpoint, a fluid may be viewed as being made up of many small interacting molecules and modeled accordingly. We developed results for one such model, the long range asymmetric ‘decomposable’ misanthrope process in Z^(n). For the long range dynamics, a particle is displaced by an amount ||d|| with probability proportional to ||d||^−(n+α). This gives rise to two different hydrodynamic equations for the evolution of the fluid density. When 0 < α < 1, we obtained an integro-partial differential equation where the integral captures the long range features of the dynamics. When α ≥ 1, we obtained a conservation law similar to Burgers’ equation; the long range nature of interactions is not apparent. For opinion dynamics, we built two one-dimensional probabilistic models of opin ion interactions. Assuming a level of attraction between all opinions, we found that a limiting behavior of consensus can often be guaranteed. The proofs regarding con sensus nicely split into multiple cases depending on the degree of attraction. We then generalized the results to higher dimensions. Lastly, we introduced the behaviors of repulsing and overshooting opinions, which can prevent consensus, and found bounds on the spread of opinions over time. This is of particular interest in one of our models, which becomes quite complex when overshooting is introduced.Type
textElectronic Dissertation
Degree Name
Ph.D.Degree Level
doctoralDegree Program
Graduate CollegeMathematics