  We study numerically statistical distributions of sums of chaotic orbit
coordinates, viewed as independent random variables, in weakly chaotic regimes
of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam
(FPU-$\beta$) oscillator chains with different boundary conditions and numbers
of particles and a microplasma of identical ions confined in a Penning trap and
repelled by mutual Coulomb interactions. For the FPU systems we show that, when
chaos is limited within "small size" phase space regions, statistical
distributions of sums of chaotic variables are well approximated for
surprisingly long times (typically up to $t\approx10^6$) by a $q$-Gaussian
($1<q<3$) distribution and tend to a Gaussian ($q=1$) for longer times, as the
orbits eventually enter into "large size" chaotic domains. However, in
agreement with other studies, we find in certain cases that the $q$-Gaussian is
not the only possible distribution that can fit the data, as our sums may be
better approximated by a different so-called "crossover" function attributed to
finite-size effects. In the case of the microplasma Hamiltonian, we make use of
these $q$-Gaussian distributions to identify two energy regimes of "weak
chaos"-one where the system melts and one where it transforms from liquid to a
gas state-by observing where the $q$-index of the distribution increases
significantly above the $q=1$ value of strong chaos.
