The Phase Transition in Statistical Models Defined on Farey Fractions

Details The Phase Transition in Statistical Models Defined on Farey Fractions The Phase Transition in Statistical Models Defined on Farey Fractions

2002-03-23

arxiv.org/abs/math-ph/0203048v3

J. Stat. Phys. (2003) 110, pp. 73-86

Physics

Mathematical Physics

14 pp., Revtex4, new version with minor errors corrected and extended sections IV and V (more detailed explanations), accepted

Scientific paper

10.1023/A:1021014627403

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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