Recent exciting experiments start to explore long-time quantum dynamics of a closed many-body system. Theoretical understanding of the dynamics is challenging due to fast entanglement generation and often relies on semi-classical approaches. In this talk, I will discuss a class of stochastic quantum many-body models called the Brownian model that is tractable away from the semi-classical limit by taking advantage of the noise average. In particular, I will show that the operator dynamics of N fermions with all-to-all stochastic interaction in 0+1D, known as the Brownian Sachdev-Ye-Kitaev model, can be mapped to the imaginary time evolution of a single spin with emergent symmetry. In higher dimensions, the noise average maps the fermion models to familiar quantum spin models, including the Heisenberg model. The mapping provides a complete picture of the dynamics from small N to large N. In the large N limit, the operator dynamics can be described by a stochastic logistic differential equation in 0+1D and FKPP (Fisher-Kolmogorov–Petrovsky–Piskunov) type equation in 1+1D.
