A numerical study of the statistics of roughness parameters for fluctuating interfaces

TL;DR

This study analyzes the statistical fluctuations of roughness parameters in self-affine interfaces through numerical simulations, emphasizing the importance of multiple samples for accurate characterization of roughness exponents and amplitudes.

Contribution

It provides a detailed statistical analysis of roughness parameters in simulated interfaces, highlighting the significance of sample-to-sample fluctuations for experimental and numerical studies.

Findings

Sample-to-sample fluctuations are large for roughness exponents.

Multiple independent realizations are necessary for reliable roughness measurements.

Fluctuations also significantly affect roughness amplitude estimates.

Abstract

Self-affine rough interfaces are ubiquitous in experimental systems, and display characteristic scaling properties as a signature of the nature of disorder in their supporting medium, i.e. of the statistical features of its heterogeneities. Different methods have been used to extract roughness information from such self-affine structures, and in particular their scaling exponents and associated prefactors. Notably, for an experimental characterization of roughness features, it is of paramount importance to properly assess sample-to-sample fluctuations of roughness parameters. Here, by performing scaling analysis based on displacement correlation functions in real and reciprocal space, we compute statistical properties of the roughness parameters. As an ideal, artifact-free reference case study and particularly targeting finite-size systems, we consider three cases of numerically simulated one-dimensional interfaces: (i) elastic lines under thermal fluctuations and free of disorder, (ii) directed polymers in equilibrium with a disordered energy landscape, and (iii) elastic lines in the critical depinning state when the external applied driving force equals the depinning force set by disorder. Our results shows that sample-to-sample fluctuations are rather large when measuring the roughness exponent. These fluctuations are also relevant for roughness amplitudes. Therefore a minimum of independent interface realizations (at least a few tens in our numerical simulations) should be used to guarantee sufficient statistical averaging, an issue often overlooked in experimental reports.