  This work proves the global stability of the Boltzmann equation (1872) with
the physical collision kernels derived by Maxwell in 1866 for the full range of
inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$, for initial
perturbations of the Maxwellian equilibrium states, as announced in
\cite{gsNonCutA}. We more generally cover collision kernels with parameters
$s\in (0,1)$ and $\gamma$ satisfying $\gamma > -n$ in arbitrary dimensions
$\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid
convergence as predicted by the celebrated Boltzmann $H$-theorem. When $\gamma
\ge -2s$, we have exponential time decay to the Maxwellian equilibrium states.
When $\gamma <-2s$, our solutions decay polynomially fast in time with any
rate. These results are completely constructive. Additionally, we prove sharp
constructive upper and lower bounds for the linearized collision operator in
terms of a geometric fractional Sobolev norm; we thus observe that a spectral
gap exists only when $\gamma \ge -2s$, as conjectured in Mouhot-Strain
\cite{MR2322149}. It will be observed that this fundamental equation, derived
by both Boltzmann and Maxwell, grants a basic example where a range of
geometric fractional derivatives occur in a physical model of the natural
world. Our methods provide a new understanding of the grazing collisions in the
Boltzmann theory.
