Thermodynamically Consistent Vibrational-Electron Heating: Generalized Model for Multi-Quantum Transitions

Accurate prediction of electron temperature ($T_{\rm e}$) is critical for non-equilibrium plasma applications ranging from hypersonic flight to plasma-assisted combustion. We recently proposed a thermodynamically consistent model for vibrational-electron heating [Phys. Fluids 37, 096141 (2025)] that enforces the convergence of $T_{\rm e}$ to the vibrational temperature ($T_{\rm v}$) at equilibrium. However, the original derivation was restricted to single-quantum transitions, limiting its validity to low-temperature regimes ($T_{\rm e} \lesssim 1.5$ 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 $\exp(-\theta_{\rm v}/T_{\rm v})$, where $\theta_{\rm v}$ is the characteristic vibrational temperature. This error exceeds 40% when $T_{\rm v}$ is greater than $\theta_{\rm v}$, effectively preventing thermal relaxation. To correct this, we derive a formulation where the total heating rate is a summation of channel-specific cooling rates $Q_{\rm e-v}^{(m)}$, each associated with a quantum jump $m$, scaled by a thermodynamic factor $\exp(m\theta_{\rm v}/T_{\rm e}-m\theta_{\rm v}/T_{\rm v})$. This generalized model preserves thermodynamic consistency by ensuring zero net energy transfer at equilibrium.