  We introduce a simple model of deterministic particles in weakly disordered
media which exhibits a transition from normal to anomalous diffusion. The model
consists of a set of non-interacting overdamped particles moving on a
disordered potential. The disordered potential can be thought as a substrate
having some "defects" scattered along a one-dimensional line. The distance
between two contiguous defects is assumed to have a heavy-tailed distribution
with a given exponent $\alpha$, which means that the defects along the
substrate are scarce if $\alpha$ is small. We prove that this system exhibits a
transition from normal to anomalous diffusion when the distribution exponent
$\alpha$ decreases, i.e., when the defects become scarcer. Thus we identify
three distinct scenarios: a normal diffusive phase for $\alpha>2$, a
superdiffusive phase for $1/2<\alpha\leq 2$, and a subdiffusive phase for
$\alpha \leq 1/2$. We also prove that the particle current is finite for all
the values of $\alpha$, which means that the transport is normal independently
of the diffusion regime (normal, subdiffusive, or superdiffusive). We give
analytical expressions for the effective diffusion coefficient for the normal
diffusive phase and analytical expressions for the diffusion exponent in the
case of anomalous diffusion. We test all these predictions by means of
numerical simulations.
