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Phase Transitions in One-Dimensional Translation Invariant Systems: A Ruelle Operator Approach

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Abstract

We consider a family of potentials \(f\), derived from the Hofbauer potentials, on the symbolic space \(\Omega =\{0,1\}^\mathbb {N}\) and the shift mapping \(\sigma \) acting on it. A Ruelle operator framework is employed to show there is a phase transition when the temperature varies in the following senses: the pressure is not analytic, there are multiple eigenprobabilities for the dual of the Ruelle operator, the DLR–Gibbs measure is not unique and finally the Thermodynamic Limit is not unique. Additionally, we explicitly calculate the critical points for these phase transitions. Some examples which are not of Hofbauer type are also considered. The non-uniqueness of the thermodynamic limit is proved by considering a version of a renewal equation. We also show that the correlations decay polynomially and compute the decay ratio.

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Authors and Affiliations

  1. Universidade de Brasilia, Brasília, DF, 70910-900, Brazil

    Leandro Cioletti

  2. UFRGS, Porto Alegre, RS, 91500, Brazil

    Artur O. Lopes

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  1. Leandro Cioletti
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  2. Artur O. Lopes
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Correspondence to Artur O. Lopes.

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Cioletti, L., Lopes, A.O. Phase Transitions in One-Dimensional Translation Invariant Systems: A Ruelle Operator Approach. J Stat Phys 159, 1424–1455 (2015). https://doi.org/10.1007/s10955-015-1202-4

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