Vibrating sandpiles: An example of systems with directionality transition

In this study, we present an oscillatory anisotropic sandpile model as an extension of the Bak, Tang, and Wiesenfeld (BTW) model. We focus on exploring the statistical properties of both local and global measures. The model introduces oscillations controlled by a time period, T, applied in the parallel direction, while the perpendicular direction follows the standard BTW model. Our results provide strong evidence of a directionality crossover at T=T^{*}: for T above (below) this value, avalanches predominantly extend along the spatial parallel direction. Notably, the spanning avalanche probability (SAP) in both directions reaches zero within an interval around T^{*}, indicating a localization of avalanches at the crossover point. SAP shows distinct behaviors on either side of this interval. Our scaling analysis reveals an additional crossover between small and large spatial scales, each characterized by unique scaling exponents. The distribution functions for avalanche mass m, size s, and duration D exhibit power-law behavior within this regime, with notable peaks at positions aligning with the crossover points; these peaks scale with T, introducing new scaling exponents. The scaling relations between mass, size, and linear perpendicular avalanche extent l_{⊥} allow us to identify and estimate the scaling exponents γ_{ml_{⊥}}=2.0±0.05 and γ_{ms}=0.95±0.03 in the small-scale regime and γ_{ml_{⊥}}=2.66±0.05 and γ_{ms}=0.68±0.03 in the large-scale regime for all T values. The mass-duration scaling exponent, γ_{mD}=1.57±0.05, remains robust against changes in T.