We introduce a generalization of the cavity, or Bethe-Peierls, method that
allows to follow Gibbs states when an external parameter, e.g. the temperature,
is adiabatically changed. This allows to obtain new quantitative results on the
static and dynamic behavior of mean field disordered systems such as models of
glassy and amorphous materials or random constraint satisfaction problems. As a
first application, we discuss the residual energy after a very slow annealing,
the behavior of out-of-equilibrium states, and demonstrate the presence of
temperature chaos in equilibrium. We also explore the energy landscape, and
identify a new transition from an computationally easier canyons-dominated
region to a harder valleys-dominated one.Comment: 6 pages, 7 figure
