Mixing, ergodicity and slow relaxation phenomena

TL;DR

This paper explores the hierarchical relationship between mixing, ergodicity, and the fluctuation-dissipation theorem in systems with memory, comparing previous diffusion studies with recent Lee recurrence relations findings, and discusses implications for slow relaxation phenomena.

Contribution

It demonstrates that the hierarchy between mixing, ergodicity, and the FDT is more general than previously thought, especially in systems like the electron gas.

Findings

Ergodicity can be violated in the dynamics of the electron gas.

The hierarchy between mixing, ergodicity, and FDT is more broadly applicable.

Implications for slow relaxation phenomena are discussed.

Abstract

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.