  This paper considers space homogenous Boltzmann kinetic equations in
dimension $d$ with Maxwell collisions (and without Grad's cut-off). An explicit
Markov coupling of the associated conservative (Nanbu) stochastic $N$-particle
system is constructed, using plain parallel coupling of isotropic random walks
on the sphere of two-body collisional directions. The resulting coupling is
almost surely decreasing, and the $L_2$-coupling creation is computed
explicitly. Some quasi-contractive and uniform in $N$ coupling / coupling
creation inequalities are then proved, relying on $2+\alpha$-moments ($\alpha
>0$) of velocity distributions; upon $N$-uniform propagation of moments of the
particle system, it yields a $N$-scalable $\alpha$-power law trend to
equilibrium. The latter are based on an original sharp inequality, which bounds
from above the coupling distance of two centered and normalized random
variables $(U,V)$ in $\R^d$, with the average square parallelogram area spanned
by $(U-U_\ast,V-V_\ast)$, $(U_\ast,V_\ast)$ denoting an independent copy. Two
counter-examples proving the necessity of the dependance on $>2$-moments and
the impossibility of strict contractivity are provided. The paper, (mostly)
self-contained, does not require any propagation of chaos property and uses
only elementary tools.
