Persistence in higher dimensions: A finite size scaling study

Area of Science:

• Statistical Physics
• Complex Systems

Background:

• Coarsening systems exhibit domain growth over time.
• Understanding the persistence of sites in these systems is crucial for characterizing their dynamics.

Purpose of the Study:

• To establish a finite-size scaling form for persistence probability in coarsening systems.
• To numerically investigate this scaling in various models and dimensions.
• To provide accurate estimates for the persistence exponent (θ) and coarsening exponent (z).

Main Methods:

• Numerical simulations of the Glauber-Ising model in dimensions d=1 to 4.
• Extension of the analysis to the diffusion problem.
• Application of finite-size scaling techniques to analyze persistence probability P(t,L).

Main Results:

• The persistence probability P(t,L) follows the finite-size scaling form L(-zθ)f(t/L(z)).
• The scaling function f(x) exhibits power-law behavior for small arguments and a constant for large arguments.
• The scaling ansatz is validated across different dimensions and models, including the diffusion problem.

Conclusions:

• The established scaling form implies a fractal distribution of persistent sites with power-law spatial correlations.
• The study provides reliable estimates for the persistence exponent (θ) and coarsening exponent (z).
• The findings offer a unified framework for understanding persistence in various coarsening phenomena.