Using detailed Monte Carlo simulations, the diffusion coefficients for relativistic runaway electron avalanches in air are found for the range of electric field strengths applicable to thunderclouds. Diffusion causes runaway electron avalanches to spread perpendicular to and parallel to the avalanche direction, resulting in much smaller peak conductivities than would be inferred otherwise. The idea that runaway electron avalanches seeded by extensive cosmic ray air showers may initiate lightning has gained considerable popularity in recent years. However, using the diffusion coefficients calculated in this paper along with the avalanche multiplication limit from X‐ray and positron feedback, it is found that even 1017 eV cosmic ray air showers do not produce high enough conductivities to significantly alter the electric field inside a thundercloud. As a result, it will be shown that at present, no compelling theoretical argument exists to suggest that cosmic ray extensive air showers initiate lightning.
