  Planar thermal equilibration is studied using direct numerical simulations of
ultracold two-dimensional (2D) ion crystals in a Penning trap with a rotating
wall. The large magnetic field of the trap splits the modes that describe
in-plane motion of the ions into two branches: High frequency cyclotron modes
dominated by kinetic energy and low frequency $\mathbf{E \times B}$ modes
dominated by potential energy associated with thermal position displacements.
Using an eigenmode analysis we extract the equilibration rate between these two
branches as a function of the ratio of the frequencies that characterize the
two branches and observe this equilibration rate to be exponentially suppressed
as the ratio increases. Under experimental conditions relevant for current work
at NIST, the predicted equilibration time is orders of magnitude longer than
any relevant experimental timescales. We also study the coupling rate
dependence on the thermal temperature and the number of ions. Besides, we show
how increasing the rotating wall strength improves crystal stability. These
details of in-plane mode dynamics help set the stage for developing strategies
to efficiently cool the in-plane modes and improve the performance of
single-plane ion crystals for quantum information processing.
