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Compared to the D-T reaction, the neutron-free proton-boron (p-$^{11}$B) fusion has garnered increasing attention in recent years. However, significant Bremsstrahlung losses pose a formidable challenge in p-$^{11}$B plasmas in achieving $Q>1$ in thermal equilibrium. The primary aim of this study is to corroborate Todd H. Rider's seminal work in the 1997 Physics of Plasmas, who investigated the feasibility of sustaining p-$^{11}$B fusion under non-thermal equilibrium conditions. Employing a series of simulations with new fusion cross-section, we assessed the minimum recirculating power that must be recycled to maintain the system's non-thermal equilibrium and found that it is substantially greater than the fusion power output, aligning with Rider's conclusions, whether under the conditions of non-Maxwellian electron distribution or Maxwellian electron distribution, reactors reliant on non-equilibrium plasmas for p-$^{11}$B fusion are unlikely to achieve net power production without the aid of highly efficient external heat engines. However, maintaining the ion temperature at 300 keV and the Coulomb logarithm at 15, while increasing the electron temperature beyond 23.33 keV set by Rider, leads to diminished electron-ion energy transfer and heightened Bremsstrahlung radiation. When the electron temperature approaches approximately 140 keV, this progression ultimately leads to a scenario where the power of Bremsstrahlung loss equals the power of electron-ion interactions, yet remains inferior to the fusion power. Consequently, this results in a net gain in energy production.

The effectiveness of multiple electron cyclotron resonance (ECR) harmonics has been thoroughly investigated in context of high current drive efficiency, generally observed in fully non-inductive operation of the low aspect ratio EXL-50 spherical tokamak (ST) powered by electron cyclotron (EC) waves. The Fokker-Plank equation is numerically solved to obtain electron distribution function, under steady state of the relativistic nonlinear Coulomb collision and quasi-linear diffusion operators, for calculating plasma current driven by the injected EC wave. For the extra-ordinary EC wave, simulation results unfold a mechanism by which electrons moving around the cold second harmonic ECR layer strongly resonate with higher harmonics via the relativistic Doppler shifted resonance condition. This feature is in fact evident above a certain value of input EC wave power in simulation, indicating it to be a non-linear phenomenon. Similar to the experimental observation, high efficiency in current drive (over 1 A/W) has indeed been found in simulation for a typical low density ($\sim 1\times10^{18}~m^{-3}$), low temperature ($\lesssim 100$ eV) plasma of EXL-50 by taking into account multi-pass absorptions in our simulation model. However, such characteristic is not found in the ordinary EC-wave study for both single-pass and multi-pass simulations, suggesting it as inefficient in driving current on our ST device., Comment: 22 pages, 15 figures. Submitted to the Nuclear Fusion journal