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This paper is concerned with approximations of the Boltzmann equation basedon the method of moments. We propose a generalization of the setting of themoment-closure problem from relative entropy to {\phi}-divergences and acorresponding closure procedure based on minimization of {\phi}-divergences.The proposed description encapsulates as special cases Grad's classical closurebased on expansion in Hermite polynomials and Levermore's entropy-basedclosure. We establish that the generalization to divergence-based closuresenables the construction of extended thermodynamic theories that avoidessential limitations of the standard moment-closure formulations such asinadmissibility of the approximate phase-space distribution, potential loss ofhyperbolicity and singularity of flux functions at local equilibrium. Thedivergence-based closure leads to a hierarchy of tractable symmetric hyperbolicsystems that retain the fundamental structural properties of the Boltzmannequation.

Moment closure approximations of the Boltzmann Equation based on {\phi}-divergences