A Model of Silence, or the Probability of "Un Ange Passe"

In French, the phrase "Un ange passe" ("An angel passes") refers to the sudden silence that falls over a co-present group -- that is, a group of people sharing the same physical space. As evidenced by the presence of similar expressions across languages and cultures, this phenomenon represents a universal feature of human conversation. At the same time, the meaning attributed to silence can differ greatly across national, cultural, and interpersonal contexts. Consequently, a wide range of studies have focused on the impact of silence on organizational productivity, its relationship to ideas and creativity, and its potential effectiveness in medical settings. Despite the important role that silence plays, very few studies have attempted to characterize its features using mathematical modeling. In this study, we propose a Markov chain model to describe the dynamics of silence in a co-present group and attempt to analyze its behavior. Our results reveal a phase-transition-like phenomenon, where the probability of silence abruptly drops to zero once individuals' awareness of the surrounding conversation falls below a critical threshold. In other words, such silence can emerge only when individuals retain a minimal degree of mutual awareness of those around them. The model proposed in this study not only offers a deeper understanding of conversational dynamics, but also holds potential for contributing to intercultural communication, organizational productivity, and medical practice.