The monatomic limit of polyatomic gases as concentration of the internal-energy measure

Abstract

In Rational Extended Thermodynamics, the monatomic limit of polyatomic gases is often singular. For the internal-state density $ϕ(I)=I^a$, with $a=(D-5)/2$, we prove that the normalized equilibrium distribution of the molecular internal energy $I$ concentrates at $I=0$ as $D\to3$, providing a rigorous justification of the Dirac prescription and its first-order correction. In classical ET14, the internal-mode energy differs from equilibrium by $-3Π/2$; positivity yields $-p<Π<(D-3)p/3$, while the exact entropy develops a boundary corner at $Π=0$. Negative dynamic pressures remain admissible, but fixed negative values do not approach the monatomic state with the prescribed total energy. This explains the need for compatible initial data. In the relativistic case, the internal energy is conditionally gamma distributed at fixed momentum; the constitutive integrals follow directly, and we derive an exact energy identity and the first correction to the Synge energy.