Phase Transitions and Topology: Exact Results

Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 33)

The preceding chapter contains a major theoretical achievement: the unbounded growth with N of certain thermodynamic observables, eventually leading to singularities in the N → limit, which are used to define the occurrence of an equilibrium phase transition, is necessarily due to appropriate topological transitions in configuration space. The relevance of topology is made especially clear by the explicit dependence of thermodynamic configurational entropy on a weighed sum of Morse indexes of configuration-space submanifolds, a relation that, loosely speaking, has some analogy with the Yang–Lee “circle theorem,” which relates thermodynamic observables to a fundamental mathematical object in the Yang–Lee theory of phase transitions: the angular distribution of the zeros of the grand-partition function on a circle in the complex fugacity plane.


Phase Transition Exact Result Euler Characteristic Betti Number Negative Eigenvalue 
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© Springer Science+Business Media 2007

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