Hydrodynamic Limits for Long Range Asymmetric Processes and Probabilistic Opinion Dynamics
PublisherThe University of Arizona.
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AbstractThis 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.
Degree ProgramGraduate College