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Generalized Number Theoretic Spin Chain-Connections to Dynamical Systems and Expectation Values

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Abstract

We generalize the number theoretic spin chain, a one-dimensional statistical model based on the Farey fractions, by introducing a parameter \(x \geqslant 0\). This allows us to write recursion relations in the length of the chain. These relations are closely related to the Lewis three-term equation, which is useful in the study of the Selberg ζ-function. We then make use of these relations and spin orientation transformations. We find a simple connection with the transfer operator of a model of intermittency in dynamical systems. In addition, we are able to calculate certain spin expectation values explicitly in terms of the free energy or correlation length. Some of these expectation values appear to be directly connected with the mechanism of the phase transition

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Author notes

  1. Jan Fiala

    Present address: Department of Physics, Clark University, Worcester, MA, 01610, USA

Authors and Affiliations

  1. Department of Physics & Astronomy, University of Maine, Orono, ME, 04469, USA

    Jan Fiala

  2. LASST and Department of Physics & Astronomy, University of Maine, Orono, ME, 04469, USA

    Peter Kleban

Authors
  1. Jan Fiala
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  2. Peter Kleban
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Correspondence to Jan Fiala.

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Fiala, J., Kleban, P. Generalized Number Theoretic Spin Chain-Connections to Dynamical Systems and Expectation Values. J Stat Phys 121, 553–577 (2005). https://doi.org/10.1007/s10955-005-7579-8

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  • DOI: https://doi.org/10.1007/s10955-005-7579-8

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