Kac model from a dynamical system's point of view

A. Péntek, Z. Toroczkai, D. H. Mayer, T. Tél

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 languageEnglish
Pages (from-to)2026-2040
Number of pages15
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume49
Issue number3
DOIs
Publication statusPublished - 1994

Fingerprint

Spin Chains
dynamical systems
Chaotic Map
Dynamical system
Analogy
Free Energy
Thermodynamics
Decay
Perron-Frobenius
Transfer Operator
One-dimensional Maps
Disordered Systems
Frustration
Long-range Interactions
free energy
Spin Systems
Finite Temperature
Decay Rate
thermodynamics
Differentiable

ASJC Scopus subject areas

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

Cite this

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

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 49, No. 3, 1994, p. 2026-2040.

Research output: Contribution to journalArticle

Péntek, A. ; Toroczkai, Z. ; Mayer, D. H. ; Tél, T. / Kac model from a dynamical system's point of view. In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 1994 ; Vol. 49, No. 3. pp. 2026-2040.
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