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Linear Glauber model.

Mário J de Oliveira1

  • 1Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, São Paulo, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 26, 2005
PubMed
Summary

We analyzed the linear Glauber model, equivalent to the voter model with noise, in d-dimensional lattices. Exact critical exponents were derived, revealing distinct behaviors in 1D versus higher dimensions.

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Area of Science:

  • Statistical Physics
  • Condensed Matter Physics
  • Complex Systems

Background:

  • The linear Glauber model and the voter model are fundamental in understanding opinion dynamics and phase transitions.
  • Investigating their properties in various dimensions provides insights into collective behavior and critical phenomena.

Purpose of the Study:

  • To determine the time-dependent and stationary properties of the linear Glauber model on a d-dimensional hypercubic lattice.
  • To derive exact critical behavior and exponents using analytical methods.
  • To establish a mapping between the Glauber model and reaction-diffusion systems.

Main Methods:

  • Utilized the Green function method for exact calculations.
  • Analyzed two-point correlations to determine critical behavior.
  • Investigated the model's behavior in d-dimensional hypercubic lattices.

Main Results:

  • Obtained exact results for two-point correlations and critical exponents.
  • For d >= 2, critical exponents are beta=0, gamma=1, nu=1/2, with logarithmic corrections at d(c)=2.
  • For d=1, critical exponents are beta=0, gamma=1/2, nu=1/2.
  • Demonstrated a mapping to a specific reaction-diffusion model.

Conclusions:

  • The linear Glauber model exhibits critical behavior with dimension-dependent exponents.
  • The Green function method provides an exact analytical approach for this model.
  • The connection to reaction-diffusion models offers alternative perspectives for analysis.

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