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Ising model in small-world networks.

Carlos P Herrero1

  • 1Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Científicas (C.S.I.C.), Campus de Cantoblanco, 28049 Madrid, Spain.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 22, 2002
PubMed
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The Ising model on small-world networks exhibits a mean-field ferromagnetic transition. Transition temperature and critical energy follow power laws with increasing disorder, with distinct heat capacity behaviors in 2D and 3D systems.

Area of Science:

  • Statistical Physics
  • Complex Networks
  • Condensed Matter Physics

Background:

  • The Ising model is a fundamental model for studying magnetism and phase transitions.
  • Small-world networks exhibit unique topological properties bridging regular lattices and random networks.
  • Understanding phase transitions in networked systems is crucial for diverse scientific fields.

Purpose of the Study:

  • To investigate the ferromagnetic phase transition of the Ising model on small-world networks.
  • To analyze the influence of network disorder (rewiring probability) on critical phenomena.
  • To characterize the thermodynamic behavior in two- and three-dimensional network structures.

Main Methods:

  • Monte Carlo simulations were employed to study the system's behavior.

Related Experiment Videos

  • The study focused on networks generated from 2D and 3D regular lattices.
  • Analysis was performed in the thermodynamic limit to observe bulk properties.

    • Main Results:

    • A mean-field character for the phase transition was observed for any finite rewiring probability (p).
    • Transition temperature and critical energy scale as power laws with respect to p for small values of disorder.
    • Heat capacity diverges logarithmically in 2D and as a power law in 3D networks as p approaches zero.

    Conclusions:

    • Small-world network topology significantly impacts the Ising model's phase transition characteristics.
    • The rewiring probability acts as a critical parameter controlling the transition's nature and scaling.
    • Distinct dimensionality-dependent behaviors in heat capacity highlight the role of network structure in critical phenomena.