Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1409.3179 · Physical review. B./Physical review. B

Local temperature of an interacting quantum system far from equilibrium

Charles Allen Stafford

Published Jun 3, 2016• 42 citations
arXivHTMLPDF

Abstract

A theory of local temperature measurement of an interacting quantum electron system far from equilibrium via a floating thermoelectric probe is developed. It is shown that the local temperature so defined is consistent with the zeroth, first, second, and third laws of thermodynamics, provided the probe-system coupling is weak and broadband. For non-broadband probes, the local temperature obeys the Clausius form of the second law and the third law exactly, but there are corrections to the zeroth and first laws that are higher order in the Sommerfeld expansion. The corrections to the zeroth and first laws are related, and can be interpreted in terms of the error of a nonideal temperature measurement. These results also hold for systems at negative absolute temperature.

DOI
10.1103/physrevb.93.245403

Related papers

Temperature and voltage measurement in quantum systems far from equilibrium

Abhay Shastry, Charles Allen Stafford

We show that a local measurement of temperature and voltage for a quantum system in steady state, arbitrarily far from equilibrium, with arbitrary interactions within the system, is unique when it exists. This is interpreted as a consequence of the second law of thermodynamics. We further derive a necessary and sufficient condition for the existence of a solution. In this regard, we find that a positive temperature solution exists whenever there is no net population inversion. However, when there is a net population inversion, we may characterize the system with a unique negative temperature. Voltage and temperature measurements are treated on an equal footing: They are simultaneously measured in a noninvasive manner, via a weakly coupled thermoelectric probe, defined by requiring vanishing charge and heat dissipation into the probe. Our results strongly suggest that a local temperature measurement without a simultaneous local voltage measurement, or vice versa, is a misleading characterization of the state of a nonequilibrium quantum electron system. These results provide a firm mathematical foundation for voltage and temperature measurements far from equilibrium.

arXiv:1603.00096•Physical review. B./Physical review. B•Oct 19, 2016•32 citations

Local temperature of out-of-equilibrium quantum electron systems

Jonathan Meair, Justin P. Bergfield, Charles Allen Stafford +1 more

We show how the local temperature of out-of-equilibrium, quantum electron systems can be consistently defined with the help of an external voltage and temperature probe. We determine sufficient conditions under which the temperature measured by the probe (i) is independent of details of the system-probe coupling, (ii) is equal to the temperature obtained from an independent current-noise measurement, (iii) satisfies the transitivity condition expressed by the zeroth law of thermodynamics, and (iv) is consistent with Carnot's theorem. This local temperature therefore characterizes the system in the common sense of equilibrium thermodynamics, but remains well defined even in out-of-equilibrium situations with no local equilibrium.

arXiv:1306.6345•Physical Review B•Jul 9, 2014•55 citations

Zeroth law in quantum thermodynamics at strong coupling: In equilibrium, not at equal temperature

Jen-Tsung Hsiang, B. L. Hu

The zeroth law of thermodynamics involves a transitivity relation (pairwise between three objects) expressed either in terms of ``equal temperature'' (ET), or ``in equilibrium'' (EQ) conditions. In conventional thermodynamics conditional on vanishingly weak system-bath coupling these two conditions are commonly regarded as equivalent. In this work we show that for thermodynamics at strong coupling they are inequivalent: namely, two systems can be in equilibrium and yet have different effective temperatures. A recent result [J.-T. Hsiang and B. L. Hu, Phys. Rev. D 103, 065001 (2021) for Gaussian quantum systems shows that an effective temperature ${T}^{*}$ can be defined at all times during a system's nonequilibrium evolution, but because of the inclusion of interaction energy, after equilibration the system's ${T}^{*}$ is slightly higher than the bath temperature ${T}_{\mathrm{B}}$, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature ${T}_{\mathrm{B}}$ will not have the same equilibrated temperature as the first object. Thus $ET\ensuremath{\ne}EQ$ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath $B$, each with a different equilibrated effective temperature. We show this is possible, and prove the existence of a generalized fluctuation-dissipation relation under this configuration. This affirms that in equilibrium is a valid and perhaps more fundamental notion which the zeroth law for quantum thermodynamics at strong coupling should be based on. Only when the system-bath coupling becomes vanishingly weak that ``temperature'' appearing in thermodynamic relations becomes universally defined and makes better physical sense.

arXiv:2012.15607•Physical review. D/Physical review. D.•Apr 13, 2021•2 citations

The breakdown of the zeroth law of thermodynamics and the definition of temperature in small quantum systems

Pei Wang

We study two small quantum systems coupled to the same reservoir which is in thermal equilibrium. By studying the particle density and the energy density in the two systems before and after they contact each other, we find that the two systems are not in thermal equilibrium with each other. Our result shows that the zeroth law of thermodynamics is broken in small quantum systems at low temperatures. Therefore, the traditional way of defining temperature fails due to the breakdown of the transitive relation of thermal equilibrium. Then we show a different way of defining temperature by attaching an auxiliary site, which plays the role of a thermometer, to the small quantum system.

arXiv:1205.4386•May 20, 2012•0 citations

The third law of thermodynamics in open quantum systems

Abhay Shastry, Yiheng Xu, Charles A. Stafford

We consider open quantum systems consisting of a finite system of independent fermions with arbitrary Hamiltonian coupled to one or more equilibrium fermion reservoirs (which need not be in equilibrium with each other). A strong form of the third law of thermodynamics, S(T) → 0 as T → 0, is proven for fully open quantum systems in thermal equilibrium with their environment, defined as systems where all states are broadened due to environmental coupling. For generic open quantum systems, it is shown that S(T) → g ln 2 as T → 0, where g is the number of localized states lying exactly at the chemical potential of the reservoir. For driven open quantum systems in a nonequilibrium steady state, it is shown that the local entropy Sx;T→0 as T(x) → 0, except for cases of measure zero arising due to localized states, where T(x) is the temperature measured by a local thermometer.

arXiv:1904.11628•The Journal of Chemical Physics•Aug 13, 2019•8 citations

Quantum Thermodynamics Without Measurements

Philipp Strasberg

Abstract We derive the basic laws of phenomenological non-equilibrium thermodynamics for small open systems, whose quantum nature can no longer be neglected. Emphasis is put from the beginning on deriving them from an underlying microscopic system on deriving them from an underlying microscopic system–bath picture. Commonly considered approximation schemes (wea k coupling master equations) are reviewed and their thermodynamics is studied. The zeroth law is discussed for small systems and exact identities for the entropy production, valid at strong coupling and in the non non-Markovian regime, are introduced. We discu ss the effect of finite baths even out of equilibrium and use the framework of repeated interactions to study microscopic non-equilibrium resources. The chapter concludes with the study of particle transport and thermoelectric devices, which were realized in experiments. This chapter focuses entirely on the dynamics of a system coupled to a bath without any external interventions.

•Jan 28, 2022•0 citations
A theory of local temperature measurement of an interacting quantum electron system far from equilibrium via a floating thermoelectric probe is developed. It is shown that the local temperature so defined is consistent with the zeroth, first, second, and third laws of thermodynamics, provided the probe-system coupling is weak and broadband. For non-broadband probes, the local temperature obeys the Clausius form of the second law and the third law exactly, but there are corrections to the zeroth and first laws that are higher order in the Sommerfeld expansion. The corrections to the zeroth and first laws are related, and can be interpreted in terms of the error of a nonideal temperature measurement. These results also hold for systems at negative absolute temperature.
We show that a local measurement of temperature and voltage for a quantum system in steady state, arbitrarily far from equilibrium, with arbitrary interactions within the system, is unique when it exists. This is interpreted as a consequence of the second law of thermodynamics. We further derive a necessary and sufficient condition for the existence of a solution. In this regard, we find that a positive temperature solution exists whenever there is no net population inversion. However, when there is a net population inversion, we may characterize the system with a unique negative temperature. Voltage and temperature measurements are treated on an equal footing: They are simultaneously measured in a noninvasive manner, via a weakly coupled thermoelectric probe, defined by requiring vanishing charge and heat dissipation into the probe. Our results strongly suggest that a local temperature measurement without a simultaneous local voltage measurement, or vice versa, is a misleading characterization of the state of a nonequilibrium quantum electron system. These results provide a firm mathematical foundation for voltage and temperature measurements far from equilibrium.
We show how the local temperature of out-of-equilibrium, quantum electron systems can be consistently defined with the help of an external voltage and temperature probe. We determine sufficient conditions under which the temperature measured by the probe (i) is independent of details of the system-probe coupling, (ii) is equal to the temperature obtained from an independent current-noise measurement, (iii) satisfies the transitivity condition expressed by the zeroth law of thermodynamics, and (iv) is consistent with Carnot's theorem. This local temperature therefore characterizes the system in the common sense of equilibrium thermodynamics, but remains well defined even in out-of-equilibrium situations with no local equilibrium.
The zeroth law of thermodynamics involves a transitivity relation (pairwise between three objects) expressed either in terms of ``equal temperature'' (ET), or ``in equilibrium'' (EQ) conditions. In conventional thermodynamics conditional on vanishingly weak system-bath coupling these two conditions are commonly regarded as equivalent. In this work we show that for thermodynamics at strong coupling they are inequivalent: namely, two systems can be in equilibrium and yet have different effective temperatures. A recent result [J.-T. Hsiang and B. L. Hu, Phys. Rev. D 103, 065001 (2021) for Gaussian quantum systems shows that an effective temperature T∗{T}^{*}T∗ can be defined at all times during a system's nonequilibrium evolution, but because of the inclusion of interaction energy, after equilibration the system's T∗{T}^{*}T∗ is slightly higher than the bath temperature TB{T}_{\mathrm{B}}TB​, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature TB{T}_{\mathrm{B}}TB​ will not have the same equilibrated temperature as the first object. Thus $ET\ensuremath{\ne}EQ$ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath BBB, each with a different equilibrated effective temperature. We show this is possible, and prove the existence of a generalized fluctuation-dissipation relation under this configuration. This affirms that in equilibrium is a valid and perhaps more fundamental notion which the zeroth law for quantum thermodynamics at strong coupling should be based on. Only when the system-bath coupling becomes vanishingly weak that ``temperature'' appearing in thermodynamic relations becomes universally defined and makes better physical sense.
We study two small quantum systems coupled to the same reservoir which is in thermal equilibrium. By studying the particle density and the energy density in the two systems before and after they contact each other, we find that the two systems are not in thermal equilibrium with each other. Our result shows that the zeroth law of thermodynamics is broken in small quantum systems at low temperatures. Therefore, the traditional way of defining temperature fails due to the breakdown of the transitive relation of thermal equilibrium. Then we show a different way of defining temperature by attaching an auxiliary site, which plays the role of a thermometer, to the small quantum system.
We consider open quantum systems consisting of a finite system of independent fermions with arbitrary Hamiltonian coupled to one or more equilibrium fermion reservoirs (which need not be in equilibrium with each other). A strong form of the third law of thermodynamics, S(T) → 0 as T → 0, is proven for fully open quantum systems in thermal equilibrium with their environment, defined as systems where all states are broadened due to environmental coupling. For generic open quantum systems, it is shown that S(T) → g ln 2 as T → 0, where g is the number of localized states lying exactly at the chemical potential of the reservoir. For driven open quantum systems in a nonequilibrium steady state, it is shown that the local entropy Sx;T→0 as T(x) → 0, except for cases of measure zero arising due to localized states, where T(x) is the temperature measured by a local thermometer.
Abstract We derive the basic laws of phenomenological non-equilibrium thermodynamics for small open systems, whose quantum nature can no longer be neglected. Emphasis is put from the beginning on deriving them from an underlying microscopic system on deriving them from an underlying microscopic system–bath picture. Commonly considered approximation schemes (wea k coupling master equations) are reviewed and their thermodynamics is studied. The zeroth law is discussed for small systems and exact identities for the entropy production, valid at strong coupling and in the non non-Markovian regime, are introduced. We discu ss the effect of finite baths even out of equilibrium and use the framework of repeated interactions to study microscopic non-equilibrium resources. The chapter concludes with the study of particle transport and thermoelectric devices, which were realized in experiments. This chapter focuses entirely on the dynamics of a system coupled to a bath without any external interventions.