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Complex systems such as glasses, gels, granular materials, and systems farfrom equilibrium exhibit violation of the ergodic hypothesis (EH) and of thefluctuation-dissipation theorem (FDT). Recent investigations in systems withmemory have established a hierarchical connection between mixing, the EH andthe FDT. They have shown that a failure of the mixing condition (MC) will leadto the subsequent failures of the EH and of the FDT. Another important point isthat such violations are not limited to complex systems: simple systems mayalso display this feature. Results from such systems are analytical andobviously easier to understand than those obtained in complex structures, wherea large number of competing phenomena are present. In this work, we review someimportant requirements for the validity of the FDT and its connection withmixing, the EH and anomalous diffusion in one-dimensional systems. We show thatwhen the FDT fails, an out-of-equilibrium system relaxes to an effectivetemperature different from that of the heat reservoir. This effectivetemperature is a signature of metastability found in many complex systems suchas spin-glasses and granular materials.Comment: to be published in the Proceedings of the XIX Sitges Conference on  ''Jammming, Yielding and Irreversible Deformation in Condensed Matter'',  M.-C.Miguel and M. Rubi eds., Lecture Notes in Physics, Springer Verlag,  Berli

Mixing, Ergodicity and the Fluctuation-Dissipation Theorem in Complex Systems