Singularity theory of smooth mappings and its applications: A survey for non-specialists

Izumiya, Shyūichi

doi:10.1142/9789812706898_0007

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2017

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“…Phase transitions, critical phenomena, and the corresponding critical exponents are fundamental topics in statistical mechanics. Though they should have a close relationship with critical points of maps [1], catastrophe theory [2,3], and singularity theory [4,5] in mathematics, there is little application of these simple physical problems. In these geometrical standpoints, it is natural to expect critical points to be singularities on a certain surface or a curve.…”

Section: Introductionmentioning

confidence: 99%

Cusp singularity in mean field Ising model

et al. 2017

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An entropy of the Ising model in the mean field approximation is derived by the Hamilton-Jacobi formalism. We consider a grand canonical ensemble with respect to the temperature and the external magnetic field. A cusp arises at the critical point, which shows a simple and new geometrical aspect of this model. In educational sense, this curve with a cusp helps students acquire a more intuitive view on statistical phase transitions.

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