Thermodynamically consistent vibrational–electron heating: Generalized model for multi-quantum transitions

Accurate prediction of electron temperature (Te) is critical for non-equilibrium plasma applications ranging from hypersonic flight to plasma-assisted combustion (PAC). We recently proposed a thermodynamically consistent model for vibrational–electron heating Rodriguez Fuentes and Parent, “Vibrational–electron heating in plasma flows: A thermodynamically consistent model,” Phys. Fluids 37, 096141 (2025) that enforces the convergence of Te to the vibrational temperature (Tv) at equilibrium. However, the original derivation was restricted to single-quantum transitions, limiting its validity to low-temperature regimes (Te≲1.5 eV). In this Letter, we generalize the model to include multi-quantum overtone transitions, extending its applicability to high-energy regimes. We demonstrate that previous models neglecting hot-band transitions incur a systematic heating error of exp(−θv/Tv), where θv is the characteristic vibrational temperature. This error exceeds 40% when Tv is greater than θv, effectively preventing thermal relaxation. To correct this, we derive a formulation where the total heating rate is a summation of channel-specific cooling rates Qe−v(m), each associated with a quantum jump m, scaled by a thermodynamic factor exp(mθv/Te−mθv/Tv). This generalized model preserves thermodynamic consistency by ensuring zero net energy transfer at equilibrium.