  Complex systems such as glasses, gels, granular materials, and systems far
from equilibrium exhibit violation of the ergodic hypothesis (EH) and of the
fluctuation-dissipation theorem (FDT). Recent investigations in systems with
memory have established a hierarchical connection between mixing, the EH and
the FDT. They have shown that a failure of the mixing condition (MC) will lead
to the subsequent failures of the EH and of the FDT. Another important point is
that such violations are not limited to complex systems: simple systems may
also display this feature. Results from such systems are analytical and
obviously easier to understand than those obtained in complex structures, where
a large number of competing phenomena are present. In this work, we review some
important requirements for the validity of the FDT and its connection with
mixing, the EH and anomalous diffusion in one-dimensional systems. We show that
when the FDT fails, an out-of-equilibrium system relaxes to an effective
temperature different from that of the heat reservoir. This effective
temperature is a signature of metastability found in many complex systems such
as spin-glasses and granular materials.

Disordered Systems and Neural Networks