  Fusion reactivity represents the integration of fusion cross-sections and the
velocity distributions of two reactants. In this study, we investigate the
upper bound of fusion reactivity for a non-thermal reactant coexisting with a
thermal Maxwellian background reactant while maintaining a constant total
energy. Our optimization approach involves fine-tuning the velocity
distribution of the non-thermal reactant. We employ both Lagrange multiplier
and Monte Carlo methods to analyze Deuterium-Tritium (D-T) and Proton-Boron11
(p-B11) fusion scenarios. Our findings demonstrate that, within the relevant
range of fusion energy, the maximum fusion reactivity can often surpass that of
the conventional Maxwellian-Maxwellian reactants case by a substantial margin,
ranging from 50\% to 300\%. These enhancements are accompanied by distinctive
distribution functions for the non-thermal reactant, characterized by one or
multiple beams. These results not only establish an upper limit for fusion
reactivity but also provide valuable insights into augmenting fusion reactivity
through non-thermal fusion, which holds particular significance in the realm of
fusion energy research.
