  The Luttinger liquid model, which describes interacting electrons in a
single-channel quantum wire, is completely integrable in the absence of
disorder and as such does not exhibit any relaxation to equilibrium. We
consider relaxation processes induced by inelastic electron-electron
interactions in a disordered Luttinger liquid, focusing on the equilibration
rate and its essential differences from the electron-electron scattering rate
as well as the rate of phase relaxation. In the first part of the paper, we
review the basic concepts in the disordered Luttinger liquid at equilibrium.
These include the elastic renormalization, dephasing, and interference-induced
localization. In the second part, we formulate a conceptually important
framework for systematically studying the nonequilibrium properties of the
strongly correlated (non-Fermi) Luttinger liquid. We derive a coupled set of
kinetic equations for the fermionic and bosonic distribution functions that
describe the evolution of the nonequilibrium Luttinger liquid. Remarkably, the
energy equilibration rate in the conducting disordered quantum wire (at
sufficiently high temperature, when the localization effects are suppressed by
dephasing) is shown to be of the order of the rate of elastic scattering off
disorder, independent of the interaction constant and temperature.
