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Critical exponents and percolation thresholds in two-dimensional systems with a finite interplane coupling.

C Thomsen1

  • 1Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 22, 2002
PubMed
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This study numerically investigates interplane coupling effects on static correlation length and critical dimension. Results show exponents reach 3D values with increasing system size, while the percolation threshold continuously transitions from 2D to 3D.

Area of Science:

  • Physics
  • Statistical Mechanics
  • Computational Physics

Background:

  • Percolation theory is crucial for understanding phase transitions in disordered systems.
  • Interplane coupling in 3D lattice models significantly influences critical phenomena.
  • Previous studies often focused on 2D systems or assumed isotropic coupling.

Purpose of the Study:

  • To numerically investigate the impact of varying interplane coupling on static correlation length exponent (nu) and critical dimension (D).
  • To determine how the percolation threshold (p(c)) transitions from 2D to 3D behavior under anisotropic coupling.
  • To validate theoretical predictions from renormalization-group theory regarding critical exponents.

Main Methods:

  • Classical site percolation simulations were employed.

Related Experiment Videos

  • Numerical studies were conducted across a range of interplane coupling strengths relative to in-plane coupling (10^-1 to 10^-6).
  • Analysis focused on determining static correlation length exponent (nu), critical dimension (D), and percolation threshold (p(c)) for different lattice structures (simple cubic, bcc, fcc).

    • Main Results:

    • Static correlation length exponent (nu) and critical dimension (D) were found to attain their three-dimensional (3D) values for sufficiently large system sizes, even with weak interplane coupling.
    • The percolation threshold (p(c)) demonstrated a continuous transition from 2D to 3D values.
    • The exponent (kappa) characterizing this transition was measured as 0.41(2), consistent across simple cubic, bcc, and fcc lattices, and matching the inverse of the susceptibility exponent (gamma = 43/18) as predicted by renormalization-group theory.

    Conclusions:

    • Interplane coupling plays a critical role in determining the dimensionality of percolation phenomena.
    • The observed transition of exponents to 3D values highlights the importance of system size in anisotropic percolation.
    • The universality of the transition exponent (kappa) across different lattice types supports renormalization-group predictions for anisotropic systems.