  Investigations on diffusion in systems with memory [I.V.L. Costa, R. Morgado,
M.V.B.T. Lima, F.A. Oliveira, Europhys. Lett. 63 (2003) 173] 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 [M.H. Lee, Phys. Rev.
B 26 (1982) 2547; M.H. Lee, Phys. Rev. Lett. 87 (2001) 250601; M.H. Lee, J.
Phys. A: Math. Gen. 39 (2006) 4651]. Lee shows that ergodicity is violated in
the dynamics of the electron gas [M.H. Lee, J. Phys. A: Math. Gen. 39 (2006)
4651]. This reinforces both works and implies that the results of [I.V.L.
Costa, R. Morgado, M.V.B.T. Lima, F.A. Oliveira, Europhys. Lett. 63 (2003) 173]
are more general than the framework in which they were obtained. Some
applications to slow relaxation phenomena are discussed.
