Topological Hypothesis on the Origin
In the previous chapter we have reported results of numerical simulations for the fluctuations of observables of a geometric nature (e.g., configurationspace curvature fluctuations) related to the Riemannian geometrization of the dynamics in configuration space.1 These quantities have been computed, using time averages, for many different models undergoing continuous phase transitions, namely φ4 lattice models with discrete and continuous symmetries and XY models. In particular, when plotted as a function of either the temperature or the energy, the fluctuations of the curvature have an apparently singular behavior at the transition point. Moreover, we have seen that the presence of a singularity in the statistical-mechanical fluctuations of the curvature at the transition point has been proved analytically for the mean-field XY model.
KeywordsPhase Transition Euler Characteristic Topological Change Singular Behavior Morse Theory
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