August 2010-July /2011
Level of Access
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 extensively studied model system, to which we are bringing new and unexpected approaches, and
2) the thermodynamics of the Farey fraction spin chain, a set of one dimensional models with interesting phase transition behavior and connections to multifractals, 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 the theory of phase transitions and has inspired several publications in number theory. The percolation research has already led to some surprising features, especially the applicability of modular forms. Ongoing work aims at extending and deepening these connections.
An important feature of the research is the relation to pure mathematics, including close collaboration with mathematicians. This has already led to some new developments in number theory in both areas mentioned, and to synergistic developments, where the number theory provides new tools to gain insight into physical systems.
We employ a variety of theoretical methods including conformal field theory, Monte Carlo simulations, mathematically exact solutions of model statistical mechanical systems, and techniques from number theory. This research extends and develops the PI's program of basic research in statistical mechanics in two dimensions and related topics.
Broader impacts of this project include the synergy with number theory already mentioned, training of graduate students in these research areas, and impacts on the PI's teaching methods. There is also ongoing significant spinoff on a very practical level, involving instrumentation for mass spectrometers (and similar instruments). This has led, in particular, to the formation of Stillwater Scientific Instruments, Inc. Our main role in this company exploits our familiarity with conformal methods.
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.
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Kleban, Peter H., "Exact Results in Model Statistical Systems" (2012). University of Maine Office of Research Administration: Grant Reports. 382.