Exact Results in Model Statistical Systems

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This award is supported by the Divisions of Materials Research, Physics, and Mathematical Sciences.Intellectual merit: This project focuses on continued research on the exact study of the statistical mechanics of model systems. The research concentrates on two areas:1) critical percolation in two dimensions, an important and very extensivelystudied model system, to which we are bringing new and unexpected approaches, and2) the thermodynamics of the Farey fraction spin chain, a set of onedimensional models with interesting phase transition behavior and connections tomultifractals, and dynamical systems.This project aims at new results and insights in both these areas.Research on the Farey models illuminates an interesting borderline case in thetheory of phase transitions and has inspired several publications in number theory.The percolation research has already led to some surprising features, especially theapplicability of modular forms. Ongoing work aims at extending and deepeningthese connections.An important feature of the research is the relation to pure mathematics,including close collaboration with mathematicians. This has already led to somenew developments in number theory in both areas mentioned, and to synergisticdevelopments, where the number theory provides new tools to gain insight intophysical systems.We employ a variety of theoretical methods including conformal fieldtheory, Monte Carlo simulations, mathematically exact solutions of modelstatistical mechanical systems, and techniques from number theory. This researchextends and develops the PI's program of basic research in statistical mechanics intwo dimensions and related topics.Broader impacts of this project include the synergy with number theoryalready mentioned, training of graduate students in these research areas, andimpacts on the PI's teaching methods. There is also ongoing significant spinoff ona very practical level, involving instrumentation for mass spectrometers (andsimilar instruments). This has led, in particular, to the formation of StillwaterScientific Instruments, Inc. Our main role in this company exploits our familiaritywith conformal methods.Non-technical Abstract:In this research funded by the Divisions of Materials Research, Physics, and Mathematical Sciences, the study of the behavior of particles and small magnets moving on surfaces will be connected to the study of pure numbers. This connection between surface physics and pure mathematics provides a fertile area for the education of students and the discovery of unexpected mathematics and physics.