New Inequalities in Equilibrium Statistical Mechanics

TL;DR

This paper reviews new thermodynamic inequalities that bound fluctuations and fidelity susceptibility in equilibrium statistical mechanics, unifying various known inequalities and deriving spectral representations.

Contribution

It introduces a unified framework for inequalities in statistical mechanics and derives spectral representations for fidelity susceptibility, showing their equivalence.

Findings

Bounds on quadratic fluctuations derived

Spectral representation of fidelity susceptibility obtained

Unified presentation of inequalities in equilibrium systems

Abstract

Recently, new thermodynamic inequalities have been obtained, which set bounds on the quadratic fluctuations of intensive observables of statistical mechanical systems in terms of the Bogoliubov - Duhamel inner product and some thermal average values. It was shown that several well-known inequalities in equilibrium statistical mechanics emerge as special cases of these results. On the basis of the spectral representation, lower and upper bounds on the one-sided fidelity susceptibility were derived in analogous terms. Here, these results are reviewed and presented in a unified manner. In addition, the spectral representation of the symmetric two-sided fidelity susceptibility is derived, and it is shown to coincide with the one-sided case. Therefore, both definitions imply the same lower and upper bounds on the fidelity susceptibility.