  We study numerically and theoretically the $d$-dimensional Hamiltonian motion
of fast particles through a field of scatterers, modeled by bounded, localized,
(time-dependent) potentials, that we refer to as (in)elastic non-dissipative
Lorentz gases. We illustrate the wide applicability of a random walk picture
previously developed for a field of scatterers with random spatial and/or
time-dependence by applying it to four other models. First, for a periodic
array of spherical scatterers in $d\geq2$, with a smooth (quasi)periodic
time-dependence, we show Fermi acceleration: the ensemble averaged kinetic
energy $\left<\|p(t)\|^2\right>$ grows as $t^{2/5}$. Nevertheless, the mean
squared displacement $\left<\|q(t)\|^2\right>\sim t^2$ behaves ballistically.
These are the same growth exponents as for random time-dependent scatterers.
Second, we show that in the soft elastic and periodic Lorentz gas, where the
particles' energy is conserved, the motion is diffusive, as in the standard
hard Lorentz gas, but with a diffusion constant that grows as $\|p_0\|^{5}$,
rather than only as $\|p_0\|$. Third, we note the above models can also be
viewed as pulsed rotors: the latter are therefore unstable in dimension $d\geq
2$. Fourth, we consider kicked rotors, and prove them, for sufficiently strong
kicks, to be unstable in all dimensions with $\left<\|p(t)\|^2\right>\sim t$
and $\left<\|q(t)\|^2\right>\sim t^3$. Finally, we analyze the singular case
$d=1$, where $\left< \|p(t)\|^2\right>$ remains bounded in time for
time-dependent non-random potentials whereas it grows at the same rate as above
in the random case.
