Avalanche dynamics in nonconservative water droplet

The self-organized criticality (SOC) exhibits excellent performance in describing the dynamic features of multi-body systems in nature. It is usually considered that conservative law is a necessary condition for the existence of SOC. However, mass or energy dissipation typically exists in the process from self-organization to a critical state in natural systems such as earthquakes and air pollution, namely, non-conservation. At present, few physical experiments have systematically explored the SOC behavior in a non-conservative system, compared to numerical simulations. This study is the first to design a non-conservative sandpile experiment using water droplets. Based on this experiment, the dynamic evolution process of avalanches in a non-conservative system has been analyzed. The results show that the avalanche size of water droplets follows a power-law distribution, as the water mass decays with time. This phenomenon obeys the rules of scale-free distribution, which is consistent with SOC behavior. Moreover, the waiting time of water droplets avalanche follows a stretched exponential distribution. These results suggest that SOC behavior still exists in non-conservative multi-body systems, which is affected by the decay coefficient. We then explore the mechanism by which the decay coefficient affects the transition of the system to a critical state in the non-conservative sandpile model (NBTW). We find that only when the decay coefficient is small do the inequality measures, Gini coefficient g and Kolkata index k, exhibit nearly universal values, similar to the mechanism of the system transitioning to a critical state in the traditional BTW model. When the decay coefficient is large, the NBTW model shows scaling-limited effects and the scaling exponent of avalanche size increases with the decay coefficient. Furthermore, we use the NBTW model to study how the decay coefficient influences the scaling exponent of avalanche size. We find that the NBTW model can effectively explain the variability of scaling exponents observed in different SOC systems and establish a quantitative relationship between the mass decay coefficient and the avalanche size scaling exponent. This study provides new insight into SOC behavior in non-conservative systems in nature.