Revisiting the Mazur bound and the Suzuki equality

TL;DR

This paper explores the relationship between the Mazur bound and Suzuki equality in quantum and classical systems, analyzing how conserved quantities influence long-time correlation bounds through analytic and numerical methods.

Contribution

It provides a detailed comparison of the Mazur bound and Suzuki equality, extending the Suzuki result to classical systems and examining conserved quantities in various models.

Findings

Suzuki equality can be extended to classical systems.

Complete sets of conserved quantities are crucial for saturating the Mazur bound.

Numerical results support the theoretical analysis across different models.

Abstract

Among the few known rigorous results for time-dependent equilibrium correlations, important for understanding transport properties, are the Mazur bound and the Suzuki equality. The Mazur inequality gives a lower bound, on the long-time average of the time-dependent auto-correlation function of observables, in terms of equilibrium correlation functions involving conserved quantities. On the other hand, Suzuki proposes an exact equality for quantum systems. In this paper, we discuss the relation between the two results and in particular, look for the analogue of the Suzuki result for classical systems. This requires us to examine as to what constitutes a complete set of conserved quantities required to saturate the Mazur bound. We present analytic arguments as well as illustrative numerical results from a number of different systems. Our examples include systems with few degrees of freedom as well as many-particle integrable models, both free and interacting.