Describing self-organized criticality as a continuous phase transition

Can the concept of self-organized criticality, exemplified by models such as the sandpile model, be described within the framework of continuous phase transitions? In this paper, we provide extensive numerical evidence supporting an affirmative answer. Specifically, we explore the Bak, Tang, and Wiesenfeld (BTW) and Manna sandpile models as instances of percolation transitions from disordered to ordered phases. To facilitate this analysis, we introduce the concept of drop density-a continuously adjustable control variable that quantifies the average number of particles added to a site. By tuning this variable, we observe a transition in the sandpile from a subcritical to a critical phase. Additionally, we define the scaled size of the largest avalanche occurring from the beginning of the sandpile as the order parameter for the self-organized critical transition and analyze its scaling behavior. Furthermore, we calculate the correlation length exponent and note its divergence as the critical point is approached. The finite-size scaling analysis of the avalanche size distribution works quite well at the critical point of the BTW sandpile.