Farey分数上统计模型的相变
The Phase Transition in Statistical Models Defined on Farey Fractions
摘要：
我们考虑几个统计模型定义的Farey分数。其中两个模型可以被看作是“自旋链”，与远程相互作用，而另一个出现在与混沌映射表现出不连续性的多重分形的研究。我们证明了这些模型都具有相同的自由能。它们的热力学行为由传递算子（Ruelle-Perron-Frobenius算子）的谱决定，该传递算子是使用生成Farey“树”的映射（表示函数）定义的。该算子的谱完全由Prellberg确定。由此可见，这些模型都有一个二级相变，比热发散的形式为[t（ln t）^2]^（-1）。自旋链模型也严格已知在相变处具有磁化的不连续性。
英文摘要：
We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as "spin chains", with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-Frobenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form [t (ln t)^2]^(-1). The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.
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