  We develop a mean-field theory of the metallic phase near the many-body
localization (MBL) transition, using the observation that a system near the MBL
transition should become an increasingly slow heat bath for its constituent
parts. As a first step, we consider the properties of a many-body localized
system coupled to a generic ergodic bath whose characteristic dynamical
timescales are much slower than those of the system. As we discuss, a wide
range of experimentally relevant systems fall into this class; we argue that
relaxation in these systems is dominated by collective many-particle
rearrangements, and compute the associated timescales and spectral broadening.
We then use the observation that the self-consistent environment of any region
in a nearly localized metal can itself be modeled as a slowly fluctuating bath
to outline a self-consistent mean-field description of the nearly localized
metal and the localization transition. In the nearly localized regime, the
spectra of local operators are highly inhomogeneous and the typical local
spectral linewidth is narrow. The local spectral linewidth is proportional to
the DC conductivity, which is small in the nearly localized regime. This
typical linewidth and the DC conductivity go to zero as the localized phase is
approached, with a scaling that we calculate, and which appears to be in good
agreement with recent experimental results.
