  One-dimensional electron systems interacting with long-range Coulomb forces
(quantum wires) show a Wigner crystal structure. We investigate in this paper
the transport properties of such a Wigner crystal in the presence of
impurities. Contrary to what happens when only short-range interactions are
included, the system is dominated by $4 k_F$ scattering on the impurities.
There are two important length scales in such a problem: one is the pinning
length above which the (quasi-)long-range order of the Wigner crystal is
destroyed by disorder. The other length $\xi_{cr}$ is the length below which
Coulomb interactions are not important and the system is behaving as a standard
Luttinger liquid with short-range interactions. We obtain the frequency and
temperature dependence of the conductivity. We show that such a system is very
similar to a classical charge density wave pinned by impurities, but with
important differences due to quantum fluctuations and long-range Coulomb
interactions. Finally we discuss our results in comparison with experimental
systems.
