Reactions which are impossible theoretically
If the $20$ J you mention is the average energy this will correspond to a certain temperature. However, molecules have a distribution of energies, caused by their random motion in solution or in the gas phase, so that some have far more energy than the average and some far less. The relative populations of vibrational and rotational energy levels a molecule has are determined by the Boltzmann (exponential) distribution. This means that at a given temperature some molecules are more likely to react than others. If the $100$ J you mention corresponds to a temperature at which the reaction proceeds with a reasonable rate constant then if you only supply $20$ J the rate constant will be $e^{-5} \approx 7.10^{-3}$ times smaller. This may be a small enough rate constant to suspect that the reaction is not proceeding, but its just a matter of waiting !

(If you have a temperature gradient across your sample then of course the reaction rate constant will also vary across the sample being greatest where the temperature is greatest.)