相变的拓扑理论
Topological Theory of Phase Transitions
摘要：
对相变的哈密顿动力学对应物的研究，结合哈密顿动力学的黎曼几何化，导致了相变的微分拓扑理论的初步形成。事实上，在对应的相变有特殊的几何变化的力学流形，发现源于其拓扑结构的变化。这些发现以及两个定理表明，可以制定超越现有理论限制的相变拓扑理论。在其他优点中，新理论适用于小$N$系统（即，在纳米和介观尺度下）中的相变，并且在没有破环的情况下。然而，该理论的初步版本是不完整的，仍然可以被反例证伪。目前的工作提供了一个相关的飞跃，导致完成发展的拓扑理论的相变铺平道路，进一步发展和应用的理论，可以不再受到阻碍。
英文摘要：
The investigation of the Hamiltonian dynamical counterpart of phase transitions, combined with the Riemannian geometrization of Hamiltonian dynamics, has led to a preliminary formulation of a differential-topological theory of phase transitions. In fact, in correspondence of a phase transition there are peculiar geometrical changes of the mechanical manifolds that are found to stem from changes of their topology. These findings, together with two theorems, have suggested that a topological theory of phase transitions can be formulated to go beyond the limits of the existing theories. Among other advantages, the new theory applies to phase transitions in small $N$ systems (that is, at nanoscopic and mesoscopic scales), and in the absence of symmetry-breaking. However, the preliminary version of the theory was incomplete and still falsifiable by counterexamples. The present work provides a relevant leap forward leading to an accomplished development of the topological theory of phase transitions paving the way to further developments and applications of the theory that can be no longer hampered.
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