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Gerald Dunne

2011-01-11
14:00:00 - 15:30:00

R102 , Astronomy and Mathematics Building

A difficult but generic problem in physics is the minimization of the thermodynamic "grand potential", in order to study the properties of a system as a function of temperature and density. For self-interacting quantum field theories, this problem can be formulated approximately as a "gap equation", which is a nonlinear functional differential equation. The Gross-Neveu model is an old, famous and well-studied self-interacting fermionic quantum field theory, but surprisingly the thermodynamic phase diagram has only recently been resolved. I show how to reduce the Gross-Neveu gap equation from a functional differential equation to a nonlinear differential equation, and show that the thermodynamics of the system is intimately related to the mKdV and AKNS integrable systems of ODEs. In higher dimensions the corresponding gap equation is related to embeddings of surfaces, and the harmonic map equations, which are also well-known in the context of integrable models.

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