### Abstract

The Kac model, a spin chain with exponentially decreasing long-range interaction, is investigated by means of a simple functional representation of the transfer operator. An analogy between the thermodynamics of spin chains and of one-dimensional (1D) chaotic maps allows us to use techniques worked out for generalized Frobenius-Perron equations to extract properties of the spin system, such as free energy and the decay rate of the correlation function. Although the Kac chain does not exhibit a phase transition, we find that the correlation decay shows a nonanalytic behavior at some finite temperature. We are also interested in a generalized version of the Kac model where the interaction still decays exponentially but in an oscillating fashion. This leads to the appearance of complicated patterns in the free energy caused by frustration which is a typical effect for disordered systems. By working out the analogy with 1D chaotic maps in more detail, we show how one can construct maps with the same thermodynamics as the spin chain. The associated maps turn out to be not smoothly differentiable, and their derivatives exhibit fractal properties.

Original language | English |
---|---|

Pages (from-to) | 2026-2040 |

Number of pages | 15 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 49 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1994 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*49*(3), 2026-2040. https://doi.org/10.1103/PhysRevE.49.2026

**Kac model from a dynamical system's point of view.** / Péntek, A.; Toroczkai, Z.; Mayer, D. H.; Tél, T.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 49, no. 3, pp. 2026-2040. https://doi.org/10.1103/PhysRevE.49.2026

}

TY - JOUR

T1 - Kac model from a dynamical system's point of view

AU - Péntek, A.

AU - Toroczkai, Z.

AU - Mayer, D. H.

AU - Tél, T.

PY - 1994

Y1 - 1994

N2 - The Kac model, a spin chain with exponentially decreasing long-range interaction, is investigated by means of a simple functional representation of the transfer operator. An analogy between the thermodynamics of spin chains and of one-dimensional (1D) chaotic maps allows us to use techniques worked out for generalized Frobenius-Perron equations to extract properties of the spin system, such as free energy and the decay rate of the correlation function. Although the Kac chain does not exhibit a phase transition, we find that the correlation decay shows a nonanalytic behavior at some finite temperature. We are also interested in a generalized version of the Kac model where the interaction still decays exponentially but in an oscillating fashion. This leads to the appearance of complicated patterns in the free energy caused by frustration which is a typical effect for disordered systems. By working out the analogy with 1D chaotic maps in more detail, we show how one can construct maps with the same thermodynamics as the spin chain. The associated maps turn out to be not smoothly differentiable, and their derivatives exhibit fractal properties.

AB - The Kac model, a spin chain with exponentially decreasing long-range interaction, is investigated by means of a simple functional representation of the transfer operator. An analogy between the thermodynamics of spin chains and of one-dimensional (1D) chaotic maps allows us to use techniques worked out for generalized Frobenius-Perron equations to extract properties of the spin system, such as free energy and the decay rate of the correlation function. Although the Kac chain does not exhibit a phase transition, we find that the correlation decay shows a nonanalytic behavior at some finite temperature. We are also interested in a generalized version of the Kac model where the interaction still decays exponentially but in an oscillating fashion. This leads to the appearance of complicated patterns in the free energy caused by frustration which is a typical effect for disordered systems. By working out the analogy with 1D chaotic maps in more detail, we show how one can construct maps with the same thermodynamics as the spin chain. The associated maps turn out to be not smoothly differentiable, and their derivatives exhibit fractal properties.

UR - http://www.scopus.com/inward/record.url?scp=35949004746&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35949004746&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.49.2026

DO - 10.1103/PhysRevE.49.2026

M3 - Article

VL - 49

SP - 2026

EP - 2040

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 3

ER -