Quantum Thermodynamic Uncertainty Relations, Generalized Current Fluctuations and Nonequilibrium Fluctuation-Dissipation Inequalities

TL;DR

This paper develops a quantum thermodynamic uncertainty relation framework based on fluctuation-dissipation inequalities, extending to classical regimes and providing improved estimates of thermodynamic uncertainties in nonequilibrium quantum systems.

Contribution

It formulates a thermal fluctuation-dissipation inequality for quantum nonequilibrium dynamics, bridging quantum and classical thermodynamic uncertainty relations with exact solutions.

Findings

Derived a thermal fluctuation-dissipation inequality valid at high temperatures.

Provided an exact expression for current-current correlations in quantum Brownian motion.

Showed the influence of bath statistics and causality on thermodynamic uncertainty relations.

Abstract

Thermodynamic uncertainty relations (TURs) represent one of the few broad-based and fundamental relations in our toolbox for tackling the thermodynamics of nonequilibrium systems. One form of TUR quantifies the minimal energetic cost of achieving a certain precision in determining a nonequilibrium current. In this initial stage of our research program, our goal is to provide the quantum theoretical basis of TURs using microphysics models of linear open quantum systems where it is possible to obtain exact solutions. In paper [Dong \textit{et al.}, Entropy {\bf 24}, 870 (2022)], we show how TURs are rooted in the quantum uncertainty principles and the fluctuation-dissipation inequalities (FDI) under fully nonequilibrium conditions. In this paper, we shift our attention from the quantum basis to the thermal manifests. Using a microscopic model for the bath's spectral density in quantum Brownian motion studies, we formulate a ``thermal'' FDI in the quantum nonequilibrium dynamics which is valid at high temperatures. This brings the quantum TURs we derive here to the classical domain and can thus be compared with some popular forms of TURs. In the thermal-energy-dominated regimes, our FDIs provide better estimates on the uncertainty of thermodynamic quantities. Our treatment includes full back-action from the environment onto the system. As a concrete example of the generalized current, we examine the energy flux or power entering the Brownian particle and find an exact expression of the corresponding current-current correlations. In so doing, we show that the statistical properties of the bath and the causality of the system+bath interaction both enter into the TURs obeyed by the thermodynamic quantities.