This work applies the moment method onto a generic form of kinetic equations, given by Boltzmann’s equation (BE), to simplify kinetic models of particle systems. This leads to a hierarchy of moment systems and the correspond- ing moment closure problem.  Grad [1] conceived of moment closure approximations  based on the expansion of the one-particle distribution in Hermite polynomials. However, Grad’s moment systems are impaired by two essential deficiencies, viz., the potential occurrence of inadmissible locally negative phase-space distributions and potential loss of hyperbolicity [2, 3]. Levermore [4] has developed a moment-closure  procedure based on constrained entropy mini- mization that leads to an exponential closure. Levermore’s moment systems retain the fundamental structural proper- ties of BE. Moreover, the moment systems form a hierarchy  of symmetric hyperbolic  systems and the corresponding distributions  are non-negative. It was later shown by Junk [5], however, that Levermore’s moment-closure procedure suffers from a realizability  problem, in that there exist moments for which the minimum-entropy distribution is non- existent. Moreover, the fluxes in Levermore’s moment systems can become arbitrarily  large in the vicinity of (local) equilibrium  when they exist on the boundary of the domain of realizable moments. Practically, the non-tractability  of Levermore’s moment-closure system poses a formidable challenge to numerical implementation.

In this work we consider alternative moment-closure relations for BE, based on non-negative approximations of the exponential function. We propose a generalization  of the setting of the moment-closure problem from Kullback- Leibler divergence [6] (i.e relative entropy) to the class of ϕ-divergences [7].  The considered ϕ-divergences con- stitute an approximation  to the Kullback-Leibler  divergence in the vicinity of (local) equilibrium. It will be shown that the approximate-exponential  closure relation can be derived via constrained minimization  of a corresponding  ϕ- divergence.  The proposed description  encapsulates as special cases Grad’s closure relation and Levermore’s entropy- based closure. Moreover,  the corresponding moment systems are symmetric  hyperbolic,  tractable and retain the fun- damental properties of BE. Finally, the opportunities pertaining to goal-oriented adaptive modeling provided by the hierarchical structure exhibited by the resulting closed systems of moment equations will be discussed.