Derive the Born's rule from environment-induced stochastic dynamics of wave-functions in an open system

Abstract: The lack of superposition of different position states or the emergence of classicality in macroscopic systems have been a puzzle for decades. Classicality exists in every measuring apparatus, and is the key for understanding what can be measured in quantum theory. Different theories have been proposed, including decoherence, einselection and the spontaneous wave-function collapse, with no consensus reached up to now. In this paper, we propose a stochastic dynamics for the wave-function in an open system (e.g. the measuring apparatus) that interacts with its environment. The trajectory of wave-function is random with a well-defined probability distribution. We show that the stochastic evolution results in the wave-function collapse and the Born's rule for specific system-environment interactions. While it reproduces the unitary evolution governed by the Schr\"{o}dinger equation when the interaction vanishes. Our results suggest that it is the way of system interacting with environment that determines whether quantum superposition dominates or classicality emerges.