拓扑和相变II。一个必要关系定理
Topology and Phase Transitions II. Theorem on a necessary relation
摘要：
本文证明了关于相变拓扑起源的一个必要性定理。我们考虑由光滑微观相互作用势V_N（q）描述的N个自由度的物理系统，以及相应的组态空间子流形族{M_v}_{v \in R}，其中M_v={q \in R^N| V_N（q）\leq v}。该定理根据流形{M_ v}_{v \in R}的Morse指数的适当加权和与热力学熵之间的解析关系，指出构型熵S^{（-）}（v）=（1/N）\log \int_{M_v} d^Nq之一的任何可能的关于N的无约束增长，即|\partial^k S^{（-）}（v）/ \partial v^k|对于k= 3，4，只能由Morse指数的加权和导出.由于这些导数之一随N的无约束增长对应于一阶或二阶相变的发生，并且由于流形的Morse指数的变化与其拓扑的变化一一对应，因此本文的主要定理指出，相变必然源于位形空间中的拓扑转变。本文给出的定理的证明不能脱离文Ⅰ的主要定理。
英文摘要：
In this second paper, we prove a necessity Theorem about the topological origin of phase transitions. We consider physical systems described by smooth microscopic interaction potentials V_N(q), among N degrees of freedom, and the associated family of configuration space submanifolds {M_v}_{v \in R}, with M_v={q \in R^N | V_N(q) \leq v}. On the basis of an analytic relationship between a suitably weighed sum of the Morse indexes of the manifolds {M_ v}_{v \in R} and thermodynamic entropy, the Theorem states that any possible unbound growth with N of one of the following derivatives of the configurational entropy S^{(-)}(v)=(1/N) \log \int_{M_v} d^Nq, that is of |\partial^k S^{(-)}(v)/\partial v^k|, for k=3,4, can be entailed only by the weighed sum of Morse indexes. Since the unbound growth with N of one of these derivatives corresponds to the occurrence of a first or of a second order phase transition, and since the variation of the Morse indexes of a manifold is in one-to-one correspondence with a change of its topology, the Main Theorem of the present paper states that a phase transition necessarily stems from a topological transition in configuration space. The proof of the Theorem given in the present paper cannot be done without Main Theorem of paper I.
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