Mixing, Ergodicity and the Fluctuation-Dissipation Theorem in complex systems

TL;DR

This paper reviews the relationships between mixing, ergodicity, and the fluctuation-dissipation theorem in complex and simple systems, highlighting how violations relate to metastability and effective temperatures.

Contribution

It elucidates the hierarchical connection between mixing, ergodicity, and the FDT, and discusses conditions for FDT validity in one-dimensional systems.

Findings

Violations of FDT indicate out-of-equilibrium states with different effective temperatures.

Analytical results demonstrate the link between mixing failure and FDT violation.

Simple systems exhibit similar violations, aiding understanding of complex systems.

Abstract

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.