Experimental Validation of a 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, α. We experimentally validate a particular case of the so-called Tsallis-Bukman scaling law, α = 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. Turbulence is one of the most complex, but ubiquitous, phenomena observed in Nature and it is related with the underlying mechanisms responsible for the micro-macro upscale causing wide-ranging effects on classical systems, like macroscopic friction in granular solids or turbulent flow regime in fluids [1][2][3][4]. The presence of multiple scales in time and space is an additional defy to a comprehensive theoretical description, and a particular effort is made in the literature to perform experiments and simulations in order to validate the proposed theoretical descriptions, particularly Tsallis nonextensive (NE) statistical mechanics [5][6][7][8].A paradigmatic work relating anomalous diffusion and turbulent-like behavior in confined granular media was presented by Radjai and Roux [4], using numerical simulations, and confirmed qualitatively by experiments by Combe and collaborators [7,8]. Radjai and Roux coined a new expression to characterize the analogies between fluctuations of particle velocities in quasistatic granular flows and the velocity fields observed in turbulent fluid flow in high Reynolds number regime, the "granulence". Most of the evidences of the granulence are based in simulations using discrete element method (DEM) but, unfortunately, one can verify a lack of quantitative experimental verification in the last years, limiting the knowledge of the micromechanics of this system based almost exclusively on numerical evidences.In the present work, we aim exactly to fill this gap by contributing with the experiment...

Investigations of formation of quasi-static vortex-structures in granular bodies using DEM

“…Its thickness was t s =10 mm (10×d 50 ). The horizontal force strongly fluctuated at the residual state [12]. There was a good qualitative agreement between DEM simulation results and real corresponding experimental outcomes [7], [13].…”

Economics and Finance: q-Statistical Stylized Features Galore

“…Before going on, let us mention that solution (13) implies that x 2 scales like t 2 3−q , hence normal diffusion for q = 1, anomalous sub-diffusion for q < 1 and super-diffusion for 1 < q < 3, which has recently been impressively validated (within a 2% experimental error) in a granular medium [10]. The important connection between the power-law nonlinear diffusion (12) and the entropy S q described here below was first established by Plastino and Plastino in [11], where they considered a more general evolution equation that reduces to (12) in the particular case of vanishing drift (i.e., F(x) = 0, ∀x).…”

Simulation and Calibration Between Parameters of Continuous Time Random Walks and Subdifusive Model

“…According to the value of the exponent α, the anomalous processes can be classified as sub-diffusive (0 < α < 1) or super-diffusive (1 < α < 2) [17]. A relation between the diffusion exponent α and the q parameter was proposed by Bukman and Tsallis [26], and recently verified experimentally [6].…”