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We study numerically and theoretically the $d$-dimensional Hamiltonian motionof fast particles through a field of scatterers, modeled by bounded, localized,(time-dependent) potentials, that we refer to as (in)elastic non-dissipativeLorentz gases. We illustrate the wide applicability of a random walk picturepreviously developed for a field of scatterers with random spatial and/ortime-dependence by applying it to four other models. First, for a periodicarray of spherical scatterers in $d\geq2$, with a smooth (quasi)periodictime-dependence, we show Fermi acceleration: the ensemble averaged kineticenergy $\left<\|p(t)\|^2\right>$ grows as $t^{2/5}$. Nevertheless, the meansquared 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 theparticles' energy is conserved, the motion is diffusive, as in the standardhard 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 beviewed as pulsed rotors: the latter are therefore unstable in dimension $d\geq2$. Fourth, we consider kicked rotors, and prove them, for sufficiently strongkicks, 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 fortime-dependent non-random potentials whereas it grows at the same rate as abovein the random case.

On (in)elastic non-dissipative Lorentz gases and the (in)stability of classical pulsed and kicked rotors