  We study how perturbations affect dynamics of integrable many-body quantum
systems, causing transition from integrability to chaos. Looking at spin
transport in the Heisenberg chain with impurities we find that in the
thermodynamic limit transport gets diffusive already at an infinitesimal
perturbation. Small extensive perturbations therefore cause an immediate
transition from integrability to chaos. Nevertheless, there is a remnant of
integrability encoded in the dependence of the diffusion constant on the
impurity density, namely, at small densities it is proportional to the square
root of the inverse density, instead of to the inverse density as would follow
from Matthiessen's rule. We show that Matthiessen's rule has to be modified in
non-ballistic systems. Results also highlight a nontrivial role of interacting
scattering on a single impurity, and that there is a regime where adding more
impurities can actually increase transport.
