Geometry, Topology and Thermodynamics
In the preceding chapter we have seen that configuration-space topology is suspected to play a significant role in the emergence of phase transition phenomena. We have summarized all the clues in the form of a working hypothesis that we called the topological hypothesis. Then this has been given strong support by a direct numerical investigation of the topological changes of configuration space of 1D and 2D lattice φ4 models. This conjecture stems from the peculiar energy density patterns of the largest Lyapunov exponent at phase transition points. In fact, Lyapunov exponents are closely related to configuration space geometry, which, in turn, can be strongly influenced by topology. However, there is another argument, independent of the Riemannian geometrization of Hamiltonian dynamics, that suggests how to make another link between Lyapunov exponents and topology.
KeywordsLyapunov Exponent Betti Number Extrinsic Curvature Morse Index Morse Function
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