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arXiv:2012.15607 · Physical review. D/Physical review. D.

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

Jen-Tsung Hsiang, B. L. Hu

Published Apr 13, 2021• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.103.085004

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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≠EQET\ensuremath{\ne}EQET\ensuremath=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.
Quantum systems strongly coupled to many-body systems equilibrate to the reduced state of a global thermal state, deviating from the local thermal state of the system as it occurs in the weak-coupling limit. Taking this insight as a starting point, we study the thermodynamics of systems strongly coupled to thermal baths. First, we provide strong-coupling corrections to the second law applicable to general systems in three of its different readings: As a statement of maximal extractable work, on heat dissipation, and bound to the Carnot efficiency. These corrections become relevant for small quantum systems and vanish in first order in the interaction strength. We then move to the question of power of heat engines, obtaining a bound on the power enhancement due to strong coupling. Our results are exemplified on the paradigmatic non-Markovian quantum Brownian motion.
A definition of nonequilibrium free energy Fs{\mathcal{F}}_{\mathrm{s}}Fs​ is proposed for dynamical Gaussian quantum open systems strongly coupled to a heat bath and the formal relation with the generating functional, the coarse-grained effective action and the influence action is indicated. For Gaussian open quantum systems exemplified by the quantum Brownian motion model studied here, a time-varying effective temperature can be introduced in a natural way, and, with it, the nonequilibrium free energy Fs{\mathcal{F}}_{\mathrm{s}}Fs​, von Neumann entropy SvN{\mathcal{S}}_{vN}SvN​ and internal energy Us{\mathcal{U}}_{\mathrm{s}}Us​ of the reduced system (SSS) can be defined accordingly. In contrast to the nonequilibrium free energy found in the literature which references the bath temperature, the nonequilibrium thermodynamic functions we find here obey the familiar relation ${\mathcal{F}}_{\mathrm{s}}(t)={\mathcal{U}}_{\mathrm{s}}(t)\ensuremath{-}{T}_{\mathrm{EFF}}(t){\mathcal{S}}_{vN}(t)$ at any and all moments of time in the system's fully nonequilibrium evolution history. After the system equilibrates they coincide, in the weak coupling limit, with their counterparts in conventional equilibrium thermodynamics. Since the effective temperature captures both the state of the system and its interaction with the bath, upon the system's equilibration, it approaches a value slightly higher than the initial bath temperature. Notably, it remains nonzero for a zero-temperature bath, signaling the existence of system-bath entanglement. Reasonably, at high bath temperatures and under ultraweak couplings, it becomes indistinguishable from the bath temperature. The nonequilibrium thermodynamic functions and relations discovered here for dynamical Gaussian quantum systems should open up useful pathways toward establishing meaningful theories of nonequilibrium quantum thermodynamics.
A model computational quantum thermodynamic network is constructed with two variable temperature baths coupled by a linker system, with an asymmetry in the coupling of the linker to the two baths. It is found in computational simulations that the baths come to "thermal equilibrium" at different bath energies and temperatures. In a sense, heat is observed to flow from cold to hot. A description is given in which a recently defined quantum entropy S_{univ}^{Q} for a pure state "universe" continues to increase after passing through the classical equilibrium point of equal temperatures, reaching a maximum at the asymmetric equilibrium. Thus, a second law account ΔS_{univ}^{Q}≥0 holds for the asymmetric quantum process. In contrast, a von Neumann entropy description fails to uphold the entropy law, with a maximum near when the two temperatures are equal, then a decrease ΔS^{vN}<0 on the way to the asymmetric equilibrium.
In a series of papers, we intend to take the perspective of open quantum systems and examine from their nonequilibrium dynamics the conditions when the physical quantities, their relations, and the laws of thermodynamics become well defined and viable for quantum many-body systems. We first describe how an open-system nonequilibrium dynamics (ONEq) approach is different from the closed combined system + environment in a global thermal state (CGTs) setup. Only after the open system equilibrates will it be amenable to conventional thermodynamics descriptions, thus quantum thermodynamics (QTD) comes at the end rather than assumed in the beginning. The linkage between the two comes from the reduced density matrix of ONEq in that stage having the same form as that of the system in the CGTs. We see the open-system approach having the advantage of dealing with nonequilibrium processes as many experiments in the near future will call for. Because it spells out the conditions of QTD's existence, it can also aid us in addressing the basic issues in quantum thermodynamics from first principles in a systematic way. We then study one broad class of open quantum systems where the full nonequilibrium dynamics can be solved exactly, that of the quantum Brownian motion of N strongly coupled harmonic oscillators, interacting strongly with a scalar-field environment. In this paper, we focus on the internal energy, heat capacity, and the third law. We show for this class of physical models, amongst other findings, the extensive property of the internal energy, the positivity of the heat capacity, and the validity of the third law from the perspective of the behavior of the heat capacity toward zero temperature. These conclusions obtained from exact solutions and quantitative analysis clearly disprove claims of negative specific heat in such systems and dispel allegations that in such systems the validity of the third law of thermodynamics relies on quantum entanglement. They are conceptually and factually unrelated issues. Entropy and entanglement will be the main theme of our second paper on this subject matter.
A system, or a set of systems, is said to be in thermal equilibrium when no net flow of heat energy occurs between its constituent parts when they are connected by a path permeable to heat (a diathermal wall).1 This condition is the endpoint of a spontaneous process: if two bodies at different temperatures are brought into thermal contact, energy will flow from the higher-temperature object to the lower-temperature object until this flow ceases and the equilibrium state is achieved.3 Within a single system, thermal equilibrium is characterized by a temperature that is both spatially uniform (the same at all points) and temporally constant (unchanging over time).4 This uniformity of macroscopic properties—such as temperature and pressure—is what defines the state of the system.1 It is important to distinguish this from the broader state of thermodynamic equilibrium. Thermal equilibrium (uniform temperature) is a necessary component, but full thermodynamic equilibrium also requires mechanical equilibrium (uniform pressure) and diffusive equilibrium (uniform chemical potential).6 Systems can achieve thermal equilibrium without achieving full thermodynamic equilibrium.6 The Zeroth Law of Thermodynamics: The "Discovery" of Temperature The concept of temperature, while intuitive, requires a formal physical principle to exist as a measurable, fundamental property. This principle is the Zeroth Law of Thermodynamics. Although it was formulated and named after the First and Second Laws, its foundational role was recognized as a logical prerequisite for the entire structure of thermodynamics.8 The law states: If system A is in thermal equilibrium with system C, and system B is in thermal equilibrium with system C, then systems A and B are in thermal equilibrium with each other.1 In this construction, system C plays the role of a thermometer.13 The profound implication of this law is mathematical: it establishes thermal equilibrium as a transitive relation 9, or more formally, an equivalence relation.14 In mathematics, any equivalence relation partitioning a set of objects implies the existence of a common, scalar property shared by all members of a given equivalence class. The Zeroth Law thus proves the existence of a scalar function of state—a property we call temperature—that is common to all systems in mutual thermal equilibrium.9 The law is, therefore, the fundamental basis for defining temperature and building consistent, reproducible temperature scales.1 This axiomatic foundation also provides the logical origin for one of temperature's most critical characteristics: it is an intensive property. An intensive property is one that does not depend on the size or amount of the system.15 If a large system 'A' is in internal thermal equilibrium 6, any two parts of it, subsystems 'A1' and 'A2', must also be in thermal equilibrium with each other. By the Zeroth Law, a thermometer 'C' that is in thermal equilibrium with the whole system 'A' must also be in thermal equilibrium with 'A1' and 'A2' individually.13 The thermometer, therefore, gives the same reading for all three. This demonstrates that the temperature of a system in thermal equilibrium is the same as the temperature of any part of it.16
Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir. For quantum thermodynamics at strong coupling, one needs to reexamine the meaning of the thermodynamic functions, the viability of the thermodynamic relations and the validity of the thermodynamic laws anew. After a brief motivation, this paper starts with a short overview of the quantum formulation based on Gelin & Thoss and Seifert. We then provide a quantum formulation of Jarzynski's two representations. We show how to construct the operator thermodynamic potentials, the expectation values of which provide the familiar thermodynamic variables. Constructing the operator thermodynamic functions and verifying or modifying their relations is a necessary first step in the establishment of a viable thermodynamics theory for quantum systems. We mention noteworthy subtleties for quantum thermodynamics at strong coupling, such as in issues related to energy and entropy, and possible ambiguities of their operator forms. We end by indicating some fruitful pathways for further developments.