  Molecular dynamics simulations are used to study different definitions of
contact at the atomic scale. The roles of temperature, adhesive interactions
and atomic structure are studied for simple geometries. An elastic, crystalline
substrate contacts a rigid, atomically flat surface or a spherical tip. The
rigid surface is formed from a commensurate or incommensurate crystal or an
amorphous solid. Spherical tips are made by bending crystalline planes or
removing material outside a sphere. In continuum theory the fraction of
atomically flat surfaces that is in contact rises sharply from zero to unity
when a load is applied. This simple behavior is surprisingly difficult to
reproduce with atomic scale definitions of contact. Due to thermal
fluctuations, the number of atoms making contact at any instant rises linearly
with load over a wide range of loads. Pressures comparable to the ideal
hardness are needed to achieve full contact at typical temperatures. A simple
harmonic mean-field theory provides a quantitative description of this behavior
and explains why the instantaneous forces on atoms have a universal exponential
form. Contact areas are also obtained by counting the number of atoms with a
time-averaged repulsive force. For adhesive interactions, the resulting area is
nearly independent of temperature and averaging interval, but usually rises
from zero to unity over a range of pressures that is comparable to the ideal
hardness. The only exception is the case of two identical commensurate
surfaces. For nonadhesive surfaces, the mean pressure is repulsive if there is
any contact during the averaging interval $\Delta t$. The associated area is
very sensitive to $\Delta t$ and grows monotonically. Similar complications are
encountered in defining contact areas for spherical tips.
