  Simple models for friction are typically one-dimensional, but real interfaces
are two-dimensional. We investigate the effects of the second dimension on
static and dynamic friction by using the Frenkel-Kontorova (FK) model. We study
the two most straightforward extensions of the FK model to two dimensions and
simulate both the static and dynamic properties. We show that the behavior of
the static friction is robust and remains similar in two dimensions for
physically reasonable parameter values. The dynamic friction, however, is
strongly influenced by the second dimension and the accompanying additional
dynamics and parameters introduced into the models. We discuss our results in
terms of the thermal equilibration and phonon dispersion relations of the
lattices, establishing a physically realistic and suitable two-dimensional
extension of the FK model. We find that the presence of additional dissipation
channels can increase the friction and produces significantly different
temperature-dependence when compared to the one-dimensional case. We also
briefly study the anisotropy of the dynamic friction and show highly nontrivial
effects, including that the friction anisotropy can lead to motion in different
directions depending on the value of the initial velocity.
