  The limit of small entropy production is reached in relaxing systems long
after preparation, and in stationary driven systems in the limit of small
driving power. Surprisingly, for extended systems this limit is not in general
the Gibbs-Boltzmann distribution, or a small departure from it. Interesting
cases in which it is not are glasses, phase-separation, and certain driven
complex fluids.
  We describe a scenario with several coexisting temperatures acting on
different timescales, and partial equilibrations at each time scale. This
scenario entails strong modifications of the fluctuation-dissipation equalities
and the existence of some unexpected reciprocity relations. Both predictions
are open to experimental verification, particularly the latter.
  The construction is consistent in general, since it can be viewed as the
breaking of a symmetry down to a residual group. It does not assume the
presence of quenched disorder. It can be -- and to a certain extent has been --
tested numerically, while some experiments are on their way. There is
furthermore the perspective that analytic arguments may be constructed to prove
or disprove its generality.
