Exponents and front fluctuations in the quenched Kardar-Parisi-Zhang universality class of one and two dimensional interfaces

Technical Deep Dive: Exponents and Front Fluctuations in the Quenched Kardar-Parisi-Zhang Universality Class

Overview

This paper presents a numerical study of the kinetic roughening properties and statistical behavior of fronts in the quenched Kardar-Parisi-Zhang (qKPZ) universality class in 1D and 2D systems. The key findings are:

• Direct computation of the full set of critical exponents ([MATH], [MATH], [MATH], and [MATH]) characterizing the surface kinetic roughening and depinning behaviors.
• Characterization of the probability density function (PDF) of front fluctuations in the growth regime, finding its asymptotic form in 1D and 2D.
• The PDF exhibits strongly non-Gaussian features, differing from the PDF of the standard KPZ equation with time-dependent noise.

Problem & Context

The quenched KPZ (qKPZ) equation describes the evolution of an interface or front in the presence of quenched (time-independent) disorder, in contrast to the standard KPZ equation which has time-dependent noise. The qKPZ equation exhibits a rich dynamical behavior with a pinning-depinning transition at a critical driving force.

At the depinning transition, the qKPZ universality class is expected to have distinct critical exponents compared to the standard KPZ class. Understanding the full set of exponents and the statistics of front fluctuations in this universality class is an important problem in the field of non-equilibrium statistical physics.

Methodology

The authors performed numerical simulations of an automaton version of the 1D and 2D qKPZ equation, with the following key aspects:

• Lattice sizes up to [MATH] for 1D and [MATH] for 2D.
• Periodic boundary conditions.
• Flat initial condition.
• Quenched disorder drawn uniformly from specific intervals.
• Calculation of observables:
  • Average front position and velocity
  • Front width (roughness)
  • Height-difference correlation function and correlation length
  • Probability density function (PDF) of front fluctuations

Results

1D Critical Exponents

• Critical force: [MATH] and velocity exponent [MATH]
• Growth exponents: [MATH], [MATH], [MATH]
• Roughness exponent: [MATH]
• Dynamic exponent: [MATH]
• The exponent values are largely compatible with the Directed Percolation Depinning universality class.

2D Critical Exponents

• Critical force: [MATH] and velocity exponent [MATH] (new result)
• Growth exponents: [MATH], [MATH], [MATH]
• Roughness exponent: [MATH]
• Dynamic exponent: [MATH]
• The exponent values show a stronger dependence on dimensionality compared to 1D.

Front Fluctuations

• The PDF of front fluctuations in the growth regime exhibits strongly non-Gaussian features, differing from the standard KPZ universality class.
• In 1D, the PDF shows an asymmetric shape with a sharp edge for negative fluctuations and a more regular tail for positive fluctuations.
• In 2D, the PDF shape is distinct from the Gumbel's first asymptotic form that describes the 2D KPZ universality class.

Interpretation

• The results demonstrate that the qKPZ universality class has distinct critical exponents and front fluctuation statistics compared to the standard KPZ class, even in the same physical dimensions.
• The non-Gaussian features of the front fluctuation PDF are a key signature of the qKPZ universality class, providing a way to unambiguously identify it.
• The findings expand our understanding of critical phenomena in non-equilibrium systems with quenched disorder, complementing previous work on the standard KPZ class.

Limitations & Uncertainties

• The results are based on numerical simulations of the discrete automaton model, which may differ quantitatively from the continuum qKPZ equation.
• The system sizes, while large, are still finite and may not fully capture the asymptotic behavior.
• The precise values of some exponents, especially in 2D, show a mild dependence on system size that is not fully resolved.

What Comes Next

• Further investigation of the connection between the qKPZ universality class and directed percolation depinning models.
• Exploration of the universal front fluctuation statistics in higher dimensions and for other quenched disorder models.
• Experimental verification of the predicted critical exponents and front fluctuation PDFs in physical systems belonging to the qKPZ class.

Sources:

• Exponents and Front Fluctuations in the Quenched Kardar-Parisi-Zhang Universality Class of One and Two Dimensional Interfaces