Persistence in q-state Potts model: a mean-field approach.

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Complex Systems

Background:

• The study focuses on the T=0 coarsening dynamics of the one-dimensional q-state Potts model.
• Persistence properties, specifically the decay of persistent spins, are crucial for understanding system evolution.
• Previous work established a power-law decay for the mean density of persistent spins, P(t) ~ t(-theta(q)).

Purpose of the Study:

• To analyze the spatial structure of the persistent region in the one-dimensional q-state Potts model.
• To derive and verify the scaling forms of correlation functions and distributions related to persistent spins.
• To investigate the influence of the persistence exponent theta(q) on the observed scaling behaviors.

Main Methods:

• Modified Mean-Field Approximation (MMFA) to study persistence properties, ignoring spatial correlations but imposing correct time dependence.
• Independent Interval Approximation (IIA) to analyze the distribution of separations between persistent spins.
• Analytical derivations of scaling functions and their limiting behaviors.
• Numerical simulations to support theoretical findings.

Main Results:

• A universal scaling form P2(r,t) = P(t)^2 * f(r/t^(1/2)) was found for the persistent site pair correlation function, independent of the persistence exponent theta(q).
• The distribution of separations between persistent spins, n(k,t), exhibits asymptotic scaling n(k,t) = t^(-2*phi) * g(t, k/t^phi), with a dynamical exponent phi = max(1/2, theta).
• Analytical results for the scaling functions' behavior at small and large separations were obtained, showing power-law and exponential decay, respectively.
• Numerical simulations confirmed the derived dynamical scaling forms and scaling function behaviors.

Conclusions:

• The study reveals universal scaling behaviors in the spatial structure of persistent regions during coarsening dynamics.
• The findings provide a deeper understanding of how persistent spins organize in one-dimensional systems.
• The results are consistent across different values of the persistence exponent theta(q) and are validated by numerical simulations.