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Polarization switching in ferroelectric hysteresis of BaTiO3 proceeds by localized nucleation and subsequent growth of domains of reverse polarization. While this process is driven by the applied electric field, thermal activation can play an important role in domain nucleation. As a consequence of the stochastic nature of thermal activation, coercive fields in small systems exhibit a significant scatter. It is demonstrated that the statistics of coercive fields observed in molecular dynamics simulations is consistent with the statistical distribution expected for field-assisted thermally activated nucleation of reverse polarization domains. The excellent quantitative agreement between the simulation data and the theory of thermally activated processes provides strong evidence that polarization switching occurs by nucleation-and-growth rather than loss of the local minimum of the Gibbs free energy function. In a pristine crystal, switching is controlled by the field at which thermal fluctuations can create a critical nucleus in the bulk (homogeneous nucleation). The analysis of crystals with various vacancy-type defects demonstrates that such defects may enable heterogeneous nucleation at reduced fields. In both cases, the statistical analysis gives access to the respective activation energies, their field dependence, and the sizes of the critical nuclei. [ABSTRACT FROM AUTHOR]