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arXiv:2206.09849 · Entropy

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

D. Reiche, Jen-Tsung Hsiang, B. L. Hu

Published Jul 23, 2022• 7 citations
arXivHTMLPDF

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 et al., Entropy 2022, 24, 870], 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.

DOI
10.3390/e24081016

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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 et al., Entropy 2022, 24, 870], 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.
Thermodynamic uncertainty relations (TURs) are a set of inequalities expressing a fundamental trade-off between precision and dissipation in nonequilibrium classical and quantum thermodynamic processes. TURs show that achieving low fluctuations in a thermodynamic quantity (e.g., heat or work) requires a minimum entropy production, with profound implications for the efficiency of biological and artificial thermodynamic processes. The accurate evaluation of TURs is a necessary requirement to quantify the fluctuation and dissipation entailed by a thermodynamic process, and ultimately to optimize the performance of quantum devices, whose operation is fundamentally a quantum thermodynamic process. Here, we simulate TURs in a transverse-field Ising model subjected to a time-dependent driving protocol, using quantum and classical computers in concert. Varying the duration and strength of the drive as well as the system size, we verify the validity of TURs, identify quantum signatures in the work statistics within the linear response regime, and observe TUR saturation in the high-temperature limit.
Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.
Thermodynamic uncertainty relations (TURs) are a set of inequalities expressing a fundamental trade-off between precision and dissipation in non-equilibrium classical and quantum thermodynamic processes. TURs show that achieving low fluctuations in a thermodynamic quantity (e.g., heat or work) requires a minimum entropy production, with profound implications for the efficiency of biological and artificial thermodynamic processes. The accurate evaluation of TURs is a necessary requirement to quantify the fluctuation and dissipation entailed by a thermodynamic process, and ultimately to optimize the performance of quantum devices, whose operation is fundamentally a quantum thermodynamic process. Here, we simulate TURs in a transverse-field Ising model subjected to a time-dependent driving protocol, using quantum and classical computers in concert. Varying the duration and strength of the drive as well as the system size, we verify the validity of TURs, identify quantum signatures in the work statistics within the linear response regime, and observe TUR saturation in the high-temperature limit.
Precision in nonequilibrium processes comes at a thermodynamic cost: suppressing fluctuations generally requires increased dissipation. Thermodynamic uncertainty relations (TURs) make this trade-off quantitative by linking current fluctuations to entropy production in classical stochastic dynamics. In the decade since its discovery, the canonical steady-state TUR and its finite-time generalizations have become a cornerstone of non-equilibrium thermodynamics, constraining the performance of molecular machines and allowing heat dissipation to be inferred from observable fluctuations. Whether the canonical TUR can be violated in a controlled quantum device remains an outstanding experimental question, in part because doing so requires resolving extremely small steady-state currents as well as their fluctuations. Here we experimentally show that steady-state quantum transport can surpass the precision permitted by the canonical TUR. We observe this violation in a superconducting quantum thermal machine coupled to a microwave waveguide acting as a cold bath and to a classical noise source providing an effective infinite-temperature bath. We observe a TUR ratio Q=1.71±0.17Q = 1.71 \pm 0.17Q=1.71±0.17, in violation of the classical bound Q≥2Q \geq 2Q≥2. Our results demonstrate a fundamental distinction between classical and quantum thermodynamics, paving the way for quantum thermal devices that achieve enhanced precision at reduced energy cost.
Thermodynamic uncertainty relations (TURs), originally discovered for classical systems, dictate the tradeoff between dissipation and fluctuations of irreversible current, specifying a minimal bound that constrains the two quantities. In a series of efforts to extend the relation to the one under more generalized conditions, it has been noticed that the bound is less tight in open quantum processes. To study the origin of the loose bounds, we consider an external field-driven transition dynamics of a two-level quantum system weakly coupled to the bosonic bath as a model of an open quantum system. The model makes it explicit that the imaginary part of quantum coherence, which contributes to dissipation to the environment, is responsible for loosening the TUR bound by suppressing the relative fluctuations in the irreversible current of transitions, whereas the real part of the coherence tightens it. Our study offers a better understanding of how quantum nature affects the TUR bound.
Thermodynamic uncertainty relations (TURs) and kinetic uncertainty relations (KURs) provide tradeoff relations between measurement fluctuation and thermodynamic cost, such as entropy production and activity. Conventionally, these relations are derived using the Cramér-Rao inequality, which involves an auxiliary perturbation in deterministic differential equations governing the time evolution of the system’s probability distribution. In this study, without relying on the previous formulation based on a deterministic evolving equation, we demonstrate that the main previously discovered uncertainty relations can be derived solely through the stochastic representation of the same dynamics. For this purpose, we propose a unified method based on stochastic representations for general Markovian dynamics. Extending beyond classical systems, we apply this method to Markovian open quantum systems by unraveling their dynamics, deriving quantum uncertainty relations that are tighter than existing ones in regimes where quantum effects play a significant role. This fully establishes uncertainty relations for both classical and quantum systems as intrinsic properties of their stochastic nature. Thermodynamic and kinetic uncertainty relations describe trade-offs between measurement fluctuations and thermodynamic costs, such as entropy production and activity. In this work, the authors present a general framework for deriving uncertainty relations in classical and quantum systems, leveraging the intrinsic stochastic properties of these systems.