Wigner-Eckart Factorization of the Polyatomic Boltzmann Collision Operator
Abstract
We extend the Wigner-Eckart factorization of the spectral Boltzmann collision operator to polyatomic gases with continuous internal energy. Because internal energies are invariant under spatial rotations, the SO(3) reduction survives the Borgnakke-Larsen energy exchange, and the twelve-dimensional collision integral collapses onto a nine-dimensional kinematic core. The core splits into a sparse geometric tensor, evaluated exactly, and a dense physical tensor, integrated by singularity-resolving Gauss rules with an auxiliary Laplace representation of the fractional energy couplings. The quadrature attains near machine precision at the fractional exponents of real gases. The collision invariants are embedded exactly, preserving the translational-internal energy exchange. The factorization compresses the operator by three to nearly four orders of magnitude and accelerates its evaluation 40-fold over dense formulations. The method is validated against the exact monatomic limit, Landau-Teller relaxation, and an analytic frozen-channel Prandtl number, and it matches a published calibration of the same kernel for N2, CO, and H2.
Related
- A fast Laguerre--Fourier spectral method for the polyatomic Boltzmann collision operator with continuous internal energy
- A Structure-Preserving Effective Linear Approximation of the Quantum Boltzmann Collision Operator
- Fock-Space Formulation of the Boltzmann Collision Operator for Maxwell Molecules
- A data-based conservative reduced collision model for the Boltzmann equation
- The Boltzmann-BGK model with nontrivial collision frequency near vacuum
BibTeX
@misc{keßler2026wignereckart,
title = {{Wigner-Eckart Factorization of the Polyatomic Boltzmann Collision Operator}},
author = {Torsten Keßler and Bas W. T. Gieling and René R. Hiemstra and Michael R. A. Abdelmalik},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29825},
eprint = {2608.29825},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29825},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wigner-Eckart Factorization of the Polyatomic Boltzmann Collision Operator
AU - Torsten Keßler
AU - Bas W. T. Gieling
AU - René R. Hiemstra
AU - Michael R. A. Abdelmalik
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29825
UR - https://arxiv.org/abs/2608.29825
AB - We extend the Wigner-Eckart factorization of the spectral Boltzmann collision operator to polyatomic gases with continuous internal energy. Because internal energies are invariant under spatial rotations, the SO(3) reduction survives the Borgnakke-Larsen energy exchange, and the twelve-dimensional collision integral collapses onto a nine-dimensional kinematic core. The core splits into a sparse geometric tensor, evaluated exactly, and a dense physical tensor, integrated by singularity-resolving Gauss rules with an auxiliary Laplace representation of the fractional energy couplings. The quadrature attains near machine precision at the fractional exponents of real gases. The collision invariants are embedded exactly, preserving the translational-internal energy exchange. The factorization compresses the operator by three to nearly four orders of magnitude and accelerates its evaluation 40-fold over dense formulations. The method is validated against the exact monatomic limit, Landau-Teller relaxation, and an analytic frozen-channel Prandtl number, and it matches a published calibration of the same kernel for N2, CO, and H2.
ER -