Two-Species Boltzmann--Kac Kinetics with Matsumoto--Yor Nonlinear Collisions: Conserved Energy and Conic Equilibria
Abstract
We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone (0, ∞)2 governed by the Matsumoto--Yor rational collision involution. We identify a non-linear conserved energy E(x,y) = x + y + 1/x and an invariant conic metric measure dμ= (dx dy)/(x(x+y)). By establishing a two-species H-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent (I(U;V)=0). In contrast to the transform-domain proof of Letac and Wesołowski~\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.
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BibTeX
@misc{itoh2026twospecies,
title = {{Two-Species Boltzmann--Kac Kinetics with Matsumoto--Yor Nonlinear Collisions: Conserved Energy and Conic Equilibria}},
author = {Yoshiaki Itoh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18892},
eprint = {2610.18892},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18892},
}
RIS
TY - UNPB
TI - Two-Species Boltzmann--Kac Kinetics with Matsumoto--Yor Nonlinear Collisions: Conserved Energy and Conic Equilibria
AU - Yoshiaki Itoh
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.18892
UR - https://arxiv.org/abs/2610.18892
AB - We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone (0, ∞)^2 governed by the Matsumoto--Yor rational collision involution. We identify a non-linear conserved energy E(x,y) = x + y + 1/x and an invariant conic metric measure dμ= (dx dy)/(x(x+y)). By establishing a two-species H-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent (I(U;V)=0). In contrast to the transform-domain proof of Letac and Wesołowski~\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.
ER -