Observable dependence of fluctuation-dissipation relations and effective temperatures

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Complex Systems

Background:

• The fluctuation-dissipation theorem (FDT) is a cornerstone of statistical mechanics, relating equilibrium properties to response functions.
• Glassy systems exhibit slow dynamics and aging, challenging standard equilibrium assumptions.
• Bouchaud's trap model provides a framework for studying aging dynamics in disordered systems.

Purpose of the Study:

• To investigate the nonequilibrium fluctuation-dissipation theorem (FDT) within the glass phase of Bouchaud's trap model.
• To derive closed-form expressions for correlation and response functions of arbitrary observables.
• To assess the applicability of FDT-derived effective temperatures in this nonequilibrium system.

Main Methods:

• Analytical derivation of correlation and response functions for an arbitrary observable 'm'.
• Analysis of the long-time limit of nonequilibrium FDT plots.
• Comparison with standard mean-field models and implications for effective temperature definitions.

Main Results:

• Closed-form correlation and response functions were obtained for arbitrary observables.
• A limiting nonequilibrium FDT plot is approached at long times for most observables.
• The FDT plot's shape depends nontrivially on the observable, with a continuously varying slope, unlike mean-field models.

Conclusions:

• Nonequilibrium FDT plots in Bouchaud's trap model cannot reliably define an effective temperature due to observable-dependent behavior.
• The findings highlight limitations in generalizing FDT-derived effective temperatures to complex, aging glassy systems.
• Further research is needed to understand the broader applicability of effective temperatures in nonequilibrium statistical mechanics.