Moment closure approximations of the Boltzmann equation based on φ-divergences

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This paper is concerned with approximations of the Boltzmann equation based on the method of moments. We propose a generalization of the setting of the moment-closure problem from relative entropy to (Formula presented.)-divergences and a corresponding closure procedure based on minimization of (Formula presented.)-divergences. The proposed description encapsulates as special cases Grad’s classical closure based on expansion in Hermite polynomials and Levermore’s entropy-based closure. We establish that the generalization to divergence-based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment-closure formulations such as inadmissibility of the approximate phase-space distribution, potential loss of hyperbolicity and singularity of flux functions at local equilibrium. The divergence-based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the Boltzmann equation.

Original languageEnglish
Pages (from-to)77-104
Number of pages28
JournalJournal of Statistical Physics
Issue number1
Publication statusPublished - Jul 2016


  • Boltzmann equation
  • Divergence
  • Entropy
  • Hyperbolic systems
  • Kinetic theory
  • Moment closure


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