Distribution function of nuclei from $e^\pm$ scattering in the presence
  of a strong primordial magnetic field
  The amplitude of the primordial magnetic field (PMF) is constrained from
observational limits on primordial nuclear abundances. Within this constraint,
it is possible that nuclear motion is regulated by Coulomb scattering with
electrons and positrons ($e^\pm$'s), while $e^\pm$'s are affected by a PMF
rather than collisions. For example, at a temperature of $10^9$ K, thermal
nuclei typically experience $\sim 10^{21}$ scatterings per second that are
dominated by very small angle scattering leading to minuscule changes in the
nuclear kinetic energy of order $\mathcal{O}$(1) eV. In this paper the upper
limit on the effects of a possible discretization of the $e^\pm$ momenta by the
PMF on the nuclear momentum distribution is estimated under the extreme
assumptions that the momentum of the $e^\pm$ is relaxed before and after
Coulomb scattering to Landau levels, and that during Coulomb scattering the PMF
is neglected. This assumption explicitly breaks the time reversal invariance of
Coulomb scattering, and the Maxwell-Boltzmann distribution is not a trivial
steady state solution of the Boltzmann equation under these assumptions. We
numerically evaluate the collision terms in the Boltzmann equation, and show
that the introduction of a special direction in the $e^\pm$ distribution by the
PMF generates no directional dependence of the collisional destruction term of
nuclei. Large anisotropies in the nuclear distribution function are then
constrained from big bang nucleosynthesis. Ultimately, we conclude that a PMF
does not significantly affect the isotropy or BBN.
