  The distribution of pairwise, relative peculiar velocities, $f(u;r)$, on
small nonlinear scales, $r$, is derived from the Press--Schechter approach.
This derivation assumes that Press--Schechter clumps are virialized and
isothermal. The virialized assumption requires that the circular velocity, $V_c
\propto M^{1/3}$, where $M$ denotes the mass of the clump. The isothermal
assumption means that the circular velocity is independent of radius. Further,
it is assumed that the velocity distribution within a clump is Maxwellian, that
the pairwise relative velocity distribution is isotropic, and that on nonlinear
scales clump-clump motions are unimportant when calculating the distribution of
velocity differences. Comparison with $N$-body simulations shows that, on small
nonlinear scales, all these assumptions are accurate. For most power spectra of
interest, the resulting line of sight, pairwise, relative velocity
distribution, $f(u_{\rm r})$, is well approximated by an exponential, rather
than a Gaussian distribution. This simple Press--Schechter model is also able
to provide a natural explanation for the observed, non-Maxwellian shape of
$f(v)$, the distribution of peculiar velocities.
