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

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