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A Geometric Perspective on Kinetic Matter-Radiation Interaction and Moment Systems

Brian K. Tran, Joshua W. Burby, Ben S. Southworth

TL;DR

The paper provides a geometric, pair-bracket formulation of kinetic matter-radiation interactions, unifying Boltzmann-type matter and radiation transport under a common Lie-Poisson and dissipative bracket framework. It introduces moment-space geometry, defines moment kernels from the underlying Hamiltonians, and presents a rigorous notion of geometric moment closure as a pullback that preserves energy and entropy. Two key applications illustrate the approach: (i) deriving a thermodynamically consistent diffusion radiation hydrodynamics model, and (ii) constructing Hamiltonian moment closures for pure radiation transport via Burby's variable moment closure framework. The work also connects the closure problem to an asymptotic, WKB-like expansion in the truncation order, and discusses avenues for data-driven, structure-preserving moment closures.

Abstract

We provide a geometric perspective on the kinetic interaction of matter and radiation, based on a pair bracket approach. We discuss the interaction of kinetic theories via dissipative brackets, with our fundamental example being the coupling of matter, described by the Boltzmann equation, and radiation, described by the radiation transport equation. We explore the transition from kinetic systems to their corresponding moment systems, provide a Hamiltonian description of such moment systems, and give a geometric interpretation of the moment closure problem for kinetic theories. As an application, we discuss in detail diffusion radiation hydrodynamics as an example of a pair bracket formulation on a space of moments corresponding to kinetic matter-radiation interaction. Additionally, using the variable moment closure framework of Burby (2023), we show how to construct Hamiltonian moment closures for kinetic transport equations with arbitrary Hamiltonian. Using this general construction, we derive novel Hamiltonian moment closures for pure radiation transport.

A Geometric Perspective on Kinetic Matter-Radiation Interaction and Moment Systems

TL;DR

The paper provides a geometric, pair-bracket formulation of kinetic matter-radiation interactions, unifying Boltzmann-type matter and radiation transport under a common Lie-Poisson and dissipative bracket framework. It introduces moment-space geometry, defines moment kernels from the underlying Hamiltonians, and presents a rigorous notion of geometric moment closure as a pullback that preserves energy and entropy. Two key applications illustrate the approach: (i) deriving a thermodynamically consistent diffusion radiation hydrodynamics model, and (ii) constructing Hamiltonian moment closures for pure radiation transport via Burby's variable moment closure framework. The work also connects the closure problem to an asymptotic, WKB-like expansion in the truncation order, and discusses avenues for data-driven, structure-preserving moment closures.

Abstract

We provide a geometric perspective on the kinetic interaction of matter and radiation, based on a pair bracket approach. We discuss the interaction of kinetic theories via dissipative brackets, with our fundamental example being the coupling of matter, described by the Boltzmann equation, and radiation, described by the radiation transport equation. We explore the transition from kinetic systems to their corresponding moment systems, provide a Hamiltonian description of such moment systems, and give a geometric interpretation of the moment closure problem for kinetic theories. As an application, we discuss in detail diffusion radiation hydrodynamics as an example of a pair bracket formulation on a space of moments corresponding to kinetic matter-radiation interaction. Additionally, using the variable moment closure framework of Burby (2023), we show how to construct Hamiltonian moment closures for kinetic transport equations with arbitrary Hamiltonian. Using this general construction, we derive novel Hamiltonian moment closures for pure radiation transport.
Paper Structure (12 sections, 5 theorems, 202 equations)

This paper contains 12 sections, 5 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Kinetic theory
  3. Hamiltonian formulation of kinetic matter
  4. Hamiltonian formulation of radiation transport
  5. Geometric Formulation of Kinetic Matter-Radiation Interactions
  6. Interaction brackets
  7. Geometric perspective on moment systems
  8. Pair bracket formulation on moments
  9. Geometric moment closure
  10. Variable moment closures for pure transport with arbitrary Hamiltonian
  11. Variable moment closures as an asymptotic expansion
  12. Conclusion

Key Result

Proposition 2.1

Assume that $G \in C(T^*Q)$ Poisson commutes with $H$, i.e., $[G,H] = 0$. Then, for a kinetic system with phase space distribution $g(t,\mathbf{x},\mathbf{p})$ and separable Hamiltonian $H(\mathbf{x},\mathbf{p}) = K(\mathbf{p}) + U(\mathbf{x})$, the Lie--Poisson evolution of the $G$-weighted kinetic where $\nabla_\mathbf{p} \nabla_\mathbf{p} H$ denotes the Hessian of $H$ with respect to $\mathbf{p

Theorems & Definitions (26)

  • Example 1.1: Fluid Moments
  • Example 1.2: Radiation Moments
  • Remark 1.1
  • Example 2.1
  • Remark 2.1
  • Definition 2.1: Kinetic Moments
  • Definition 2.2: Weighted Kinetic Moments
  • Definition 2.3: Symmetric Tensor Product
  • Proposition 2.1
  • proof
  • ...and 16 more