  Noise-induced phase transitions are common in various complex systems, from
physics to biology. In this article, we investigate the emergence of crucial
events in noise-induced phase transition processes and their potential
significance for understanding complexity in such systems. We utilize the
first-passage time technique and coordinate transformations to study the
dynamics of the system and identify crucial events. Furthermore, we employ
Diffusion Entropy Analysis, a powerful statistical tool, to characterize the
complexity of the system and quantify the information content of the identified
events. Our results show that the emergence of crucial events is closely
related to the complexity of the system and can provide insight into its
behavior. This approach may have applications in diverse fields, such as
climate modeling, financial markets, and biological systems, where
understanding the emergence of crucial events is of great importance.
