0 30 50 32 v 1 1 5 M ay 2 00 3 Topology and Phase Transitions : Theorem on a necessary relation
0 30 50 32 v 1 1 5 M ay 2 00 3 Topology and Phase Transitions : Theorem on a necessary relation
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0 30 50 32 v 1 1 5 May 2 00 3 拓扑和相变：必要关系定理
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
L. Spinelli
中科院分区：
文献类型：
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作者：
R. Franzosi;M. Pettini;L. Spinelli
我们证明了一个定理，该定理为一级或二级相变的发生建立了必要的拓扑条件；为了使这些发生，配置空间的某些子流形的拓扑必须在相变点发生变化。该定理适用于以下范围内的一系列光滑、有限范围和限制势 V，描述了限制在坐标连续变化的有限空间区域中的系统。相关的配置空间子流形是势函数V的水平集{Σv := V −1 N (v)}v∈R，N是自由度数，v是势能。证明继续表明，在任意 v 值区间内等位超曲面 {Σv}v∈R 的微分同胚性假设下，在相应的温度区间，赫尔莫兹自由能在 N 中一致收敛到其热力学极限，至少在二次可微函数类内。
We prove a theorem that establishes a necessary topological condition for the occurrence of first or second order phase transitions; in order for these to occur, the topology of certain submanifolds of configuration space must necessarily change at the phase transition point. The theorem applies to a wide class of smooth, finite-range and confining potentials V bounded below, describing systems confined in finite regions of space with continuously varying coordinates. The relevant configuration space submanifolds are the level sets {Σv := V −1 N (v)}v∈R of the potential function V , N is the number of degrees of freedom and v is the potential energy. The proof proceeds by showing that, under the assumption of diffeomorphicity of the equipotential hypersurfaces {Σv}v∈R in an arbitrary interval of values for v, the Helmoltz free energy is uniformly convergent in N to its thermodynamic limit, at least within the class of twice differentiable functions, in the corresponding interval of temperature.