1 0 O ct 2 00 6 Mixing , Ergodicity and slow relaxation phenomena

Investigations on diffusion in systems with memory [1] have established a hierarchical connection between mixing, ergodicity, and the fluctuation-dissipation theorem (FDT). This hierarchy means that ergodicity is a necessary condition for the validity of the FDT, and mixing is a necessary condition for ergodicity. In this work, we compare those results with recent investigations using the Lee recurrence relations method [2–4]. Lee shows that ergodicity is violated in the dynamics of the electron gas [4]. This reinforces both works and implies that the results of [1] are more general than the framework in which they were obtained. Some applications to slow relaxation phenomena are discussed. Introduction – More than a hundred years after its formulation by Boltzmann, the ergodic hypothesis (EH) still drives the attention of the mathematics and physics community. Many situations have been found in which the EH is broken [1, 3–5]. On the other hand, the mixing condition (MC), despite being omnipresent in relaxation processes, has seldomly been the object of investigation [1, 6]. Recent studies on anomalous diffusion [1, 7] have established a strong hierarchy, which in growing generality is: the FDT, EH, and the MC. Since the FDT is a less general " theorem " , we shall focus only on the EH and the MC.