Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
复制标题
势能景观鞍点引起的熵的非解析性
DOI：
10.1088/1742-5468/2008/04/p04025
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
S. Schreiber
中科院分区：
文献类型：
--
作者：
M. Kastner;O. Schnetz;S. Schreiber
研究了经典多粒子系统势的鞍点与其玻尔兹曼熵的解析性质之间的关系。对于有限的系统，每个鞍点被发现导致非解析性的玻尔兹曼熵，和这个非解析项的函数形式推导出具有莫尔斯属性的潜在的一般情况下。随着系统尺寸的增加，非解析项的阶数无限增长，导致熵的可微性增加。尽管如此，鞍点数量无限增长的分布可能会导致热力学极限的相变。分析了热力学极限下鞍点对态密度的贡献，导出了相变发生所需的鞍点分布及其曲率条件。有了这些结果，在球形模型中的拓扑签名的令人困惑的缺乏得到了阐明。作为进一步的应用，平均场XY模型和平均场k-三角模型的相变被证明是由曲率消失的鞍点引起的。
The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its Boltzmann entropy is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived for the generic case of potentials having the Morse property. With increasing system size the order of the nonanalytic term grows unboundedly, leading to an increasing differentiability of the entropy. Nonetheless, a distribution of an unboundedly growing number of saddle points may cause a phase transition in the thermodynamic limit. Analysing the contribution of the saddle points to the density of states in the thermodynamic limit, conditions on the distribution of saddle points and their curvatures are derived which are necessary for a phase transition to occur. With these results, the puzzling absence of topological signatures in the spherical model is elucidated. As further applications, the phase transitions of the mean-field XY model and the mean-field k-trigonometric model are shown to be induced by saddle points of vanishing curvature.