We establish a Navier-Stokes-Fourier limit for solutions of the Boltzmann equation considered over any periodic spatial domain of dimension two or more. We do this for a broad class of collision kernels that relaxes the Grad small deflection cutoff condition for hard potentials made by Golse and Saint-Raymond, and includes for the first time kernel arising from soft potentials. All appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that are compact. Every limit point of such a family is governed by a Leray weak solution of a Navier-Stokes-Fourier system for all time. Key tools include the relative entropy cutoff method of Saint Raymond, the L^1 velocity averaging result of Golse and Saint Raymond, and some new estimates.
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