Experimental validation of nonextensive scaling law in confined granular media

Abstract: In this letter, we address the relationship between the statistical fluctuations of grain displacements for a full quasistatic plane shear experiment, and the corresponding anomalous diffusion exponent, $\alpha$. We experimentally validate a particular case of the so-called Tsallis-Bukman scaling law, $\alpha = 2 / (3 - q)$, where $q$ is obtained by fitting the probability density function (PDF) of the measured fluctuations with a $q$-Gaussian distribution, and the diffusion exponent is measured independently during the experiment. Applying an original technique, we are able to evince a transition from an anomalous diffusion regime to a Brownian behavior as a function of the length of the strain-window used to calculate the displacements of grains in experiments. The outstanding conformity of fitting curves to a massive amount of experimental data shows a clear broadening of the fluctuation PDFs as the length of the strain-window decreases, and an increment in the value of the diffusion exponent - anomalous diffusion. Regardless of the size of the strain-window considered in the measurements, we show that the Tsallis-Bukman scaling law remains valid, which is the first experimental verification of this relationship for a classical system at different diffusion regimes. We also note that the spatial correlations show marked similarities to the turbulence in fluids, a promising indication that this type of analysis can be used to explore the origins of the macroscopic friction in confined granular materials.

Experimental Validation of Nonextensive Scaling Law in Confined Granular Media

The paper "Experimental Validation of Nonextensive Scaling Law in Confined Granular Media" presents an empirical study that investigates the connection between statistical fluctuations in grain displacements within a quasistatic plane shear experiment and the anomalous diffusion behavior characterized by the diffusion exponent, denoted as $\alpha$. This research notably confirms the validity of the Tsallis-Bukman scaling law, given by $\alpha = 2 / (3 - q)$, for a classical system across different diffusion regimes, through fitting the probability density function (PDF) of measured fluctuations with a $q$-Gaussian distribution.

The experimental setup involved a quasistatic simple shear test utilizing a $1\gamma2\varepsilon$ apparatus, simulating a confined granular medium formed by wooden rollers with varying diameters. The granular system's kinematic behavior was meticulously captured using Digital Image Correlation (DIC), yielding a high-resolution analysis of grain dynamics.

Significantly, the study identifies a transition from an anomalous diffusion regime to a Brownian motion as a function of the strain-window length, bringing to light the expected broadening of the PDFs of displacement fluctuations with a decrease in strain-window size. Such a transition is elucidated through the $q$-Gaussian value's obtainment at different strain-windows and its relationship to the diffusion exponent $\alpha$. The experimental results consistently adhere to the Tsallis-Bukman scaling law, even under varying diffusion conditions, thus contributing an unprecedented verification of this theoretical relationship for nonextensive systems.

Furthermore, the experiment uncovers parallels between granular media and fluid turbulence, notably in spatial correlations. The emergence of intricate vortex structures within the granular displacement field mirrors turbulent eddies found in fluid dynamics, suggesting potential insights into macroscopic friction mechanisms within granular assemblies.

The implications of this research extend to both practical and theoretical domains. Practically, this work enables a finer comprehension of the micro-mechanical behaviors influencing macroscopic properties such as friction in confined granular systems. Theoretically, it bolsters nonextensive statistical mechanics as a framework for interpreting complex systems characterized by strong correlations and long-range interactions. Future work might explore the implications of these findings for the modeling and prediction of behavior in granular flow and other complex systems, potentially enhancing computational simulations with empirical validation from nonextensive frameworks. This advancement stands to influence subsequent studies on granular flow mechanics, contributing to broader applications in engineering and material sciences.