  We investigate the origin of diffusion in non-chaotic systems. As an example,
we consider 1-$d$ map models whose slope is everywhere 1 (therefore the
Lyapunov exponent is zero) but with random quenched discontinuities and
quasi-periodic forcing. The models are constructed as non-chaotic
approximations of chaotic maps showing deterministic diffusion, and represent
one-dimensional versions of a Lorentz gas with polygonal obstacles (e.g., the
Ehrenfest wind tree model). In particular, a simple construction shows that
these maps define non-chaotic billiards in space-time. The models exhibit, in a
wide range of the parameters, the same diffusive behavior of the corresponding
chaotic versions. We present evidence of two sufficient ingredients for
diffusive behavior in one-dimensional, non-chaotic systems: i) a finite-size,
algebraic instability mechanism, and ii) a mechanism that suppresses periodic
orbits.
