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REVIEW 4 major objections 5 minor 115 references

The fourth law of thermodynamics: steepest entropy ascent

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Pith's one-line read This paper proposes a fourth law of thermodynamics: every nonequilibrium system's irreversible evolution is steepest entropy ascent under a state-dependent metric.

desk verdict Beretta's 'fourth law' is a competent synthesis of an existing family of gradient-flow models, but with the metric and timescale left free it states a representation theorem rather than a falsifiable law. read the letter →

arxiv 1908.05768 v2 pith:ARMBS33G submitted 2019-08-15 cond-mat.stat-mech quant-ph

Gian Paolo Beretta This is my paper · ORCID
classification cond-mat.stat-mechquant-ph PACS 05.70.-a05.70.Ln
keywords steepestentropyascentfourthlawofthermodynamicsnonequilibriumOnsagerreciprocityfluctuation-dissipationtheoremrate-controlledconstrained-equilibriumdissipativedynamicsproduction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that all dissipative nonequilibrium dynamics share one variational principle: entropy increases along the steepest ascent direction compatible with conservation laws, with steepness measured by a metric that varies from system to system. The author proposes to rank this rule alongside the first three laws of thermodynamics. As evidence, the paper points to the convergence of many modeling traditions on the same steepest-entropy-ascent structure, and shows that in the rate-controlled constrained-equilibrium approximation the rule reproduces and extends Onsager reciprocity and fluctuation-dissipation relations to far-from-equilibrium states. If the claim holds, the job of modeling a nonequilibrium system becomes the job of finding its metric and its intrinsic dissipation time.

What carries the argument

The load-bearing object is a Riemannian metric field $G_{\gamma}$ on state space together with a state-dependent time scale $\tau_{\gamma}$, the intrinsic dissipation time. The dissipative tangent vector is $\Pi = \tau_{\gamma}^{-1} G_{\gamma}^{-1}$ applied to the component of the entropy gradient orthogonal to the conserved charges; the metric converts the entropy differential into a preferred direction, and $\tau_{\gamma}$ sets how fast the state moves along it. From this one expression the paper builds the generalized conductivity matrix $L_{jk}(\gamma)$ as a Gram matrix, so its symmetry and non-negative definiteness follow automatically from the properties of the metric.

What would settle it

If a well-characterized system with a well-defined entropy is found whose far-from-equilibrium force-flux relations give an asymmetric generalized conductivity matrix that cannot be symmetrized by any choice of state-dependent metric, or whose measured relaxation path is incompatible with every metric steepest-ascent flow, the universal fourth law would be refuted.

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Extended reading notes

Core claim

The central claim is Rule 4: for every nonequilibrium state for which entropy is well defined, the dissipative component of the time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints, as measured by a local metric field. The paper's constructive result is that, within the rate-controlled constrained-equilibrium (quasi-equilibrium) approximation, the SEA evolution yields a nonlinear generalization of Onsager reciprocity: the dissipative rate of each constraint is a combination of the constraint affinities through a symmetric, positive-semidefinite matrix of generalized conductivities computed as a Gram matrix of the projected constraint derivatives. Thus reciprocity and fluctuation-dissipation structure survive far from equilibrium in a quasi-linear force-flux form.

Load-bearing premise

The law applies only to states for which entropy is well defined, and the paper's load-bearing premise is that operational entropy can be defined for every nonequilibrium state, including local subsystems; the extension to correlated states is still an open debate.

Editorial extensions

If this is right

  • Near-equilibrium Onsager reciprocity becomes the linearized special case of a far-from-equilibrium quasi-linear structure.
  • The same Gram-matrix construction yields a far-nonequilibrium generalization of the fluctuation-dissipation theorem.
  • Systems with identical state spaces and conserved charges differ in dynamics only through their metric field and intrinsic dissipation time.
  • Under the SEA evolution equation, maximum-entropy states are the only stable equilibrium states, so part of the second law emerges as a theorem.
  • Coarse-graining relations between levels of description must include a rule connecting the SEA metrics at the two levels.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the fourth law is right, inverse modeling of nonequilibrium systems becomes a metric-recovery problem: relaxation data determine $G_{\gamma}$ and $\tau_{\gamma}$, which may make model reduction more systematic.
  • A directly testable prediction is that far-from-equilibrium force-flux data should exhibit a symmetric generalized conductivity matrix even when the force-flux curves are nonlinear.
  • The law's scope hinges on the contested allocation of entropy among correlated subsystems, so its sharpest future test may be in strongly correlated quantum systems rather than dilute gases or classical liquids.
  • The variational form suggests a numerical consistency check: any proposed dissipative update can be tested for whether some positive-definite metric makes it a steepest-ascent step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a 'fourth law of thermodynamics': every nonequilibrium state for which entropy is well-defined is equipped with a metric in state space such that the irreversible component of its time evolution is in the direction of steepest entropy ascent (SEA) compatible with conservation constraints. The paper argues that the SEA structure has emerged independently in many frameworks (GENERIC, metriplectic, gradient flows, chemical kinetics, quantum thermodynamics) and derives, under the rate-controlled constrained-equilibrium (RCCE) approximation, nonlinear force-flux relations and generalized Onsager reciprocity. It closes by addressing a referee's objection concerning stochastic thermodynamics and by restating the law as Rule (4) in the Conclusion.

Significance. If true, this would be a major claim: a universal variational principle for dissipative dynamics, unifying diverse formalisms and extending Onsager reciprocity far from equilibrium. The paper is explicit about its core equations and engages a broad literature, and it is honest about contested points, including the status of entropy for correlated states and the referee's objection. However, as formulated, the law appears to have no falsifiable empirical content: the metric G_γ and the intrinsic dissipation time τ_γ are free, state-dependent fields, so the SEA form can represent any charge-conserving, entropy-producing dissipative dynamics. The RCCE derivation then yields quasi-linear force-flux relations that are identities built from the definitions. These issues, rather than the admitted domain limitations, are decisive for the assessment.

major comments (4)
  1. [Section 4, Eq. (13)] Eq. (13) defines Π_γ = (1/τ_γ) G_γ^{-1} (δS/δγ)|_C with G_γ and τ_γ free, state-dependent fields. At each fixed state, for any charge-conserving dissipative vector v with positive entropy production, there exists a symmetric positive definite G_γ satisfying G_γ v = (δS/δγ)|_C (choose coordinates with (δS/δγ)|_C = e^1 and set the first column of G_γ to e^1 with a positive-definite completion); τ_γ then rescales v. Thus Eq. (13) imposes no restriction beyond conservation and dS/dt ≥ 0, and the 'fourth law' as stated is not falsifiable. Section 1 and Section 4 explicitly embrace the freedom of G_γ ('varies from system to system'), confirming that the law is a representation theorem, not a substantive dynamical constraint.
  2. [Section 5, Eqs. (19)–(22)] The generalized Onsager conductivities L_jk are defined in Eq. (19) as (1/τ_γ)(δa_j|_C | G_γ^{-1} | δa_k|_C), so the symmetry and non-negative definiteness that the text derives from the metric are present by construction. The force-flux relations Π_{Ak} = Σ_j χ_j L_jk in Eq. (22) and the quadratic entropy production in Eq. (21) are therefore identities following from the definition of L_jk and Eq. (16), not independent predictions. To claim an extension of Onsager reciprocity, the paper would need to show that the L_jk so defined coincide with the transport matrix measured or computed from the underlying kinetics; no independent identification is provided.
  3. [Section 4 / Footnote 4] The law is restricted to states 'for which entropy is well-defined,' and Footnote 4 concedes that the extension to correlated states of interacting systems is 'still the subject of intense debate' because correlation entropy cannot be uniquely allocated to subsystems. Since the Conclusion's Rule (4) explicitly covers local subsystems, the paper's universality claim is narrower than stated. This is an acknowledged limitation, but it further reduces the domain in which the proposed law could be tested.
  4. [Sections 3–4] The claimed convergence of many nonequilibrium frameworks is asserted rather than demonstrated: the equivalence of SEA with GENERIC and metriplectic structures is delegated to Refs. [65,87], and the referee's stochastic-thermodynamics objection is answered in the Conclusion by consistency arguments rather than by showing that the SEA form actually holds in a concrete stochastic model with negative entropy-production fluctuations. For a claim at the level of a law of Nature, this leaves the inductive evidence largely uncritical.
minor comments (5)
  1. [Front matter and captions] The paper mixes Italian and English front matter ('Sommario', 'Figura', 'Riferimenti bibliografici'); these should be translated for an English-language journal.
  2. [Figures 2 and 3] The captions refer to panels (a) and (b) that are not clearly labeled in the reproduced figures; the figure files should match the captions.
  3. [Section 2] The notation for state vectors (γγγ) is typeset as repeated characters and is sometimes ambiguous; a consistent bold math notation would improve readability.
  4. [Section 5] The phrase 'As shown in [107,65]' precedes a result that is in fact derived in the text; the sentence should distinguish the present derivation from previous work.
  5. [Section 5, after Eq. (21)] The sentence 'The natural properties ... grant automatically' would be more precise as 'imply'; the intended linear-algebra statement is standard but should be stated cleanly.

Circularity Check

2 steps flagged · score 8.0 of 10

The 'fourth law' is definitional: free metric and time fields make the SEA form impose no constraint, and the Section 5 Onsager reciprocity is a Gram-matrix identity.

  1. self definitional [Section 4, Eq. (13); Section 1, final paragraph (metric field definition); Section 4 text on G_γ and τ_γ]
    "For every state γ of a system ... the component ... responsible for entropy generation (dissipation) is determined by a local non-degenerate metric operator G_γγγ and a local characteristic time τ_γγγ. ... The functional dependence of the SEA metric on the state variables varies from system to system and is in fact what characterizes its nonequilibrium behavior."

    At a fixed state, Eq. (13) sets Π = (1/τ) G^{-1} (δS/δγ)|_C. Since G and τ are free state-dependent fields, the map (G, τ) → (1/τ) G^{-1} ω, with ω = (δS/δγ)|_C, is surjective onto the open half-space {v : ω(v) > 0}: choosing coordinates with ω = e_1, any such v is obtained from a symmetric positive-definite G^{-1} whose first column is proportional to v. Hence the 'fourth law' imposes no restriction beyond charge conservation and positive entropy production, which are already assumed, and the dissipative component is defined to be steepest entropy ascent rather than predicted to be.

  2. self definitional [Section 5, Eqs. (19)-(22) and following paragraph]
    "By defining the 'RCCE nonequilibrium Onsager generalized conductivities' L_jk(γ) = (1/τ)(δa_j/δγ|_C | G^{-1}_γ | δa_k/δγ|_C) ... The natural properties of symmetry and positive definiteness of the non-degenerate metric G_γγγ grant automatically (no additional assumptions needed) its invertibility (G^{-1}_γγγ) and the symmetry and non-negative definiteness of matrix L_jk."

    Eq. (19) defines L_jk as a Gram matrix of G^{-1}. Symmetry and non-negative definiteness are properties of every Gram matrix, so the 'far-nonequilibrium Onsager reciprocity' in Eqs. (21)-(22) is not an independent consequence of the fourth law but an immediate consequence of the definition. Because G is arbitrary, any quasi-linear force-flux relation with symmetric positive-semidefinite coefficients can be represented in this form; the claimed extension of Onsager reciprocity and fluctuation-dissipation relations has no falsifiable content beyond the definitional choice of the metric.

full rationale

The central claim fails as an empirical law because Eq. (13) defines the irreversible component through the free fields G_γ and τ_γ. The paper itself says the metric's functional dependence 'varies from system to system and is in fact what characterizes its nonequilibrium behavior,' and Figure 4 illustrates that different materials with identical entropy landscapes differ only by the metric. At each state, once the entropy gradient ω=(δS/δγ)|_C is fixed, the map (G,τ) → (1/τ)G^{-1}ω is surjective onto {v : ω(v)>0}; hence every charge-conserving, entropy-producing vector field is steepest entropy ascent for some metric. The only constraints are the already-assumed conservation and positive entropy production, so the fourth law adds no falsifiable restriction. Section 5's far-from-equilibrium Onsager reciprocity is likewise definitional: Eq. (19) defines L_jk as a Gram matrix of G^{-1}, whose symmetry and non-negative definiteness are automatic, so the quasi-linear force-flux structure is a renaming of the metric structure rather than a derived prediction. Self-citations to [65,87,107] establish equivalences and formulas, but the core reduction is already visible in the paper's own definitions; no external falsification is involved.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper's central derivation rests on postulating a metric field G_gamma and a dissipation time tau_gamma, which are free, system-specific parameters. The RCCE approximation is an additional modeling assumption. The universal-law claim itself is an axiom in the argument, supported by citation rather than derivation. No new physical entities are introduced.

free parameters (3)
  • Metric operator field G_gamma = not specified (system-specific)
    The fourth law postulates a local non-degenerate metric at every state; its functional form determines the direction of dissipation and is not fixed by the theory. It is a free input that characterizes the system's nonequilibrium behavior (Section 4).
  • Intrinsic dissipation time tau_gamma = not specified (system-specific)
    The scalar field sets the speed of entropy ascent; like the metric, it is system-specific and not derived from the first three laws (Section 4).
  • Choice of RCCE constraints (slow variables) = chosen by modeler
    The far-from-equilibrium Onsager extension in Section 5 depends on a selected set of rate-controlling constraints; different choices yield different L_jk, so this is a free modeling choice.
assumptions (5)
  • domain assumption Entropy is well-defined for every nonequilibrium state, including local subsystems.
    Required by the statement of the fourth law; the paper cites its own operational definition [42,5,6] but footnote 4 notes extension to correlated states is debated (Section 4, footnote 4).
  • ad hoc to paper State space is a Riemannian manifold with a non-degenerate metric G_gamma.
    The metric field is the central postulate of the law, not a consequence of other principles (Section 4).
  • domain assumption Dynamics separates into reversible and dissipative components, with dissipative components orthogonal to conserved charges.
    Assumed in Eqs. (9)-(11); standard in SEA/GENERIC, but not derived here.
  • domain assumption RCCE approximation: the state always lies on the maximum-entropy manifold subject to slow constraints.
    Assumed in Eq. (5) and used for the Onsager extension in Section 5.
  • ad hoc to paper The diverse nonequilibrium frameworks cited have indeed converged to SEA structure.
    This is the paper's main claim; it is supported by citation to prior work and examples, not by a systematic proof (Sections 1 and 6).

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Cite this review

Pith. "Pith review of The fourth law of thermodynamics: steepest entropy ascent." pith.science (2026). https://pith.science/paper/ARMBS33G

@misc{pith2026190805768,
  author       = {Pith},
  title        = {Pith review of: The fourth law of thermodynamics: steepest entropy ascent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARMBS33G}},
  note         = {Machine review of arXiv:1908.05768}
}
read the original abstract

When thermodynamics is understood as the science (or art) of constructing effective models of natural phenomena by choosing a minimal level of description capable of capturing the essential features of the physical reality of interest, the scientific community has identified a set of general rules that the model must incorporate if it aspires to be consistent with the body of known experimental evidence. Some of these rules are believed to be so general that we think of them as laws of Nature, such as the great conservation principles, whose "greatness" derives from their generality, as masterfully explained by Feynman in one of his legendary lectures. The second law of thermodynamics is universally contemplated among the great laws of Nature. In this paper we show that, in the past four decades, an enormous body of scientific research devoted to modeling the essential features of nonequilibrium natural phenomena has converged from many different directions and frameworks towards the general recognition (albeit still expressed in different but equivalent forms and language) that another rule is also indispensable and reveals another great law of Nature that we propose to call the \caporali{fourth law of thermodynamics}. We state it as follows: every nonequilibrium state of a system or local subsystem for which entropy is well-defined must be equipped with a metric in state space with respect to which the irreversible component of its time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints. To illustrate the power of the fourth law, we derive (nonlinear) extensions of Onsager reciprocity and fluctuation-dissipation relations to the far-nonequilibrium realm within the framework of the rate-controlled constrained-equilibrium (RCCE) approximation (also known as the quasi-equilibrium approximation).

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Reference graph

Works this paper leans on

115 extracted references · 42 canonical work pages

  1. [1]

    The Character of Physical Law

    Feynman RP. 1964 The great conservation principles. In the Messenger Lectures series titled “The Character of Physical Law” given at Cornell University by Richard Feynman and recorded by BBC. London: BBC

    1964

  2. [2]

    1950 The nature of physical reality: A philosophy of modern physics

    Margenau H. 1950 The nature of physical reality: A philosophy of modern physics. New York, NY: McGraw Hill

    1950

  3. [3]

    2005Thermodynamics: Foundations and applications

    Gyftopoulos EP, Beretta GP. 2005Thermodynamics: Foundations and applications. Mineola, NY: Dover Publications

  4. [4]

    1976 A unified quantum theory of mechanics and thermodynamics

    Hatsopoulos GN, Gyftopoulos EP. 1976 A unified quantum theory of mechanics and thermodynamics. Part IIa. Available energy.Found. Phys. 6 127–141. doi:10.1007/BF00708955

    doi:10.1007/bf00708955 1976

  5. [5]

    2014 Recent progress in the definition of thermodynamic entropy.Entropy 16 1547–1570

    Zanchini E, Beretta GP. 2014 Recent progress in the definition of thermodynamic entropy.Entropy 16 1547–1570. doi:10.3390/e16031547

    doi:10.3390/e16031547 2014

  6. [6]

    Beretta GP, Zanchini E. 2019 New definitions of thermodynamic temperature and entropy not based on the concepts of heat and thermal reservoir.Atti della Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche Matematiche e Naturali. 97 (Suppl.1) A1. doi:10.1478/AAPP.97S1A1

    doi:10.1478/aapp.97s1a1 2019

  7. [7]

    1998 General projection operator formalism for the dynamics and thermodynamics of complex fluids.Phys

    Öttinger HC. 1998 General projection operator formalism for the dynamics and thermodynamics of complex fluids.Phys. Rev. E 57 1416. doi:10.1103/PhysRevE.57.1416

    doi:10.1103/physreve.57.1416 1998

  8. [8]

    2014 Contact geometry of mesoscopic thermodynamics and dynamics.Entropy 161652-1686

    Grmela M. 2014 Contact geometry of mesoscopic thermodynamics and dynamics.Entropy 161652-1686. doi:10.3390/e16031652

    doi:10.3390/e16031652 2014

Show all 115 references
  1. [9]

    2014 Time reversal in nonequilibrium thermodynamics.Phys

    Pavelka M, Klika V, Grmela M. 2014 Time reversal in nonequilibrium thermodynamics.Phys. Rev. E 90 062131. doi:10.1103/PhysRevE.90.062131

    2014 doi

  2. [10]

    2018 Coarse-graining via the fluctuation-dissipation theorem and large-deviation theory

    Montefusco A, Peletier MA, Öttinger HC. 2018 Coarse-graining via the fluctuation-dissipation theorem and large-deviation theory. arXiv:1809.07253

    2018 arXiv

  3. [11]

    2007 Comparison of invariant manifolds for model reduction in chemical kinetics.Commun

    Chiavazzo E, Gorban AN, Karlin IV. 2007 Comparison of invariant manifolds for model reduction in chemical kinetics.Commun. Comput. Phys.2 964–992

    2007

  4. [12]

    2010 Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics

    Lebiedz D, Reinhardt V, Siehr J. 2010 Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics. J. Comput. Phys. 229 6512–6533. doi:10.1016/j.jcp.2010.05.008

    2010 doi

  5. [13]

    Hiremath V, S. B. Pope SB. 2013 A study of the rate-controlled constrained-equilibrium dimen- sion reduction method and its different implementations. Combust. Theory Model. 17 260–293. doi:10.1080/13647830.2012.752109 12

    2013

  6. [14]

    2018 Systematic constraint selection strategy for rate-controlled constrained-equilibrium modeling of complex nonequilibrium chemical kinetics

    Beretta GP, Rivadossi L, Janbozorgi M. 2018 Systematic constraint selection strategy for rate-controlled constrained-equilibrium modeling of complex nonequilibrium chemical kinetics. J. Non-Equilib. Thermodyn. 43 121–130. doi:10.1515/jnet-2017-0055

    2018 doi

  7. [15]

    1976 A unified quantum theory of mechanics and thermodynamics

    Hatsopoulos GN, Gyftopoulos EP. 1976 A unified quantum theory of mechanics and thermodynamics. Part I. Postulates.Found. Phys. 6 15–31. doi:10.1007/BF00708660

    1976 doi

  8. [16]

    1976 A unified quantum theory of mechanics and thermodynamics

    Hatsopoulos GN, Gyftopoulos EP. 1976 A unified quantum theory of mechanics and thermodynamics. Part IIb. Stable equilibrium states.Found. Phys. 6 439–455. doi:10.1007/BF00715033

    1976 doi

  9. [17]

    1976 A unified quantum theory of mechanics and thermodynamics

    Hatsopoulos GN, Gyftopoulos EP. 1976 A unified quantum theory of mechanics and thermodynamics. Part III. Irreducible quantal dispersions.Found. Phys. 6 561–570. doi:10.1007/BF00715108

    1976 doi

  10. [18]

    1999 The physics and mathematics of the second law of thermodynamics.Phys

    Lieb EH, Yngvason J. 1999 The physics and mathematics of the second law of thermodynamics.Phys. Reps. 310 1–96. doi:10.1016/S0370-1573(98)00082-9

    1999 doi

  11. [19]

    2013 The entropy concept for nonequilibrium states.Proc

    Lieb EH, Yngvason J. 2013 The entropy concept for nonequilibrium states.Proc. Royal Society A469 20139408. doi:10.1098/rspa.2013.0408

    2013

  12. [20]

    2014 Entropy meters and the entropy of non-extensive systems.Proc

    Lieb EH, Yngvason J. 2014 Entropy meters and the entropy of non-extensive systems.Proc. Royal Society A 470 20140192.doi:10.1098/rspa.2014.0192

    2014

  13. [21]

    2015 What is the second law? ASME Journal of Energy Resources Technology 137 021003

    Gyftopoulos EP, Beretta GP. 2015 What is the second law? ASME Journal of Energy Resources Technology 137 021003. doi:10.1115/1.4026379

    2015 doi

  14. [22]

    2015 The second laws of quantum thermodynamics

    Brandao F, Horodecki M, Ng N, Oppenheim J, Wehner S. 2015 The second laws of quantum thermodynamics. PNAS 112 3275–3279. doi:10.1073/pnas.1411728112

    2015 doi

  15. [23]

    2016 Axiomatic relation between thermodynamic and information-theoretic entropies.Phys

    Weilenmann M, Krämer L, Faist P, Renner R. 2016 Axiomatic relation between thermodynamic and information-theoretic entropies.Phys. Rev. Lett.117 260601. doi:10.1103/PhysRevLett.117.260601

    2016 doi

  16. [24]

    2008 Where is the entropy challenge?AIP Conf

    Hatsopoulos GN, Beretta GP. 2008 Where is the entropy challenge?AIP Conf. Proc. Series 1033 34–54. doi:10.1063/1.2979057

    2008 doi

  17. [25]

    1965Principles of general thermodynamics

    Hatsopoulos GN, Keenan JH. 1965Principles of general thermodynamics. New York: Wiley

  18. [26]

    1979Thermionic Energy Conversion, Vol

    Hatsopoulos GN, Gyftopoulos EP. 1979Thermionic Energy Conversion, Vol. 2, Theory, Technology, and Application. Cambridge, MA: MIT Press

  19. [27]

    2005Thermodynamics: Fundamentals for applications

    O’Connell JP, Haile JM. 2005Thermodynamics: Fundamentals for applications. Cambridge University Press

  20. [28]

    2005 Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems

    Ebeling W, Sokolov IM. 2005 Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Singapore: World Scientific

    2005

  21. [29]

    2008Understanding non-equilibrium thermodynamics

    Lebon G, Jou D, Casas-Vázquez J. 2008Understanding non-equilibrium thermodynamics. Foundations, applications, frontiers. (Vol. 295). Berlin: Springer

  22. [30]

    2008Non-equilibrium thermodynamics of heterogeneous systems

    Kjelstrup S, Bedeaux D. 2008Non-equilibrium thermodynamics of heterogeneous systems. Singapore: World Scientific

  23. [31]

    2015 What is heat?ASME Journal of Energy Resources Technology137 021006

    Beretta GP, Gyftopoulos EP. 2015 What is heat?ASME Journal of Energy Resources Technology137 021006. http://dx.doi.org/10.1115/1.4026382

    2015 doi

  24. [32]

    2008 Local effective dynamics of quantum systems: A generalized approach to work and heat.EPL, Europhys

    Weimer H, Henrich MJ, Rempp F, Schröder H, Mahler G. 2008 Local effective dynamics of quantum systems: A generalized approach to work and heat.EPL, Europhys. Lett.83 30008. doi:10.1209/0295- 5075/83/30008

    2008 doi

  25. [33]

    2012 Quantum refrigerators and the third law of thermodynamics.Phys

    Levy A, Alicki R, Kosloff R. 2012 Quantum refrigerators and the third law of thermodynamics.Phys. Rev. E 85 061126. doi:10.1103/PhysRevE.85.061126 13

    2012 doi

  26. [34]

    2014 Work extraction and thermodynamics for individual quantum systems

    Skrzypczyk P, Short AJ, Popescu S. 2014 Work extraction and thermodynamics for individual quantum systems. Nature Commun. 5 4185. doi: 10.1038/ncomms5185

    2014 doi

  27. [35]

    2015 Quantum thermodynamics of general quantum processes

    Binder F, Vinjanampathy S, Modi K, Goold J. 2015 Quantum thermodynamics of general quantum processes. Phys. Rev. E 91 032119. doi 10.1103/PhysRevE.91.032119

    2015 doi

  28. [36]

    2015 Nature of heat in strongly coupled open quantum systems

    Esposito M, Ochoa MA, Galperin M. 2015 Nature of heat in strongly coupled open quantum systems. Phys. Rev. B92 235440. doi:10.1103/PhysRevB.92.235440

    2015 doi

  29. [37]

    2017 Resource theory for work and heat.Phys

    Sparaciari C, Oppenheim J, Fritz T. 2017 Resource theory for work and heat.Phys. Rev. A96 052112. doi:10.1103/PhysRevA.96.052112

    2017 doi

  30. [38]

    2013 Resource theory of quantum states out of thermal equilibrium.Phys

    Brandao FG, Horodecki M, Oppenheim J, Renes JM, Spekkens RW. 2013 Resource theory of quantum states out of thermal equilibrium.Phys. Rev. Lett111 250404. 10.1103/PhysRevLett.111.250404

    2013 doi

  31. [39]

    2016 Beyond heat baths: Generalized resource theories for small-scale thermodynamics

    Yunger-Halpern N, Renes JM. 2016 Beyond heat baths: Generalized resource theories for small-scale thermodynamics. Phys. Rev. E 93 022126. doi:10.1103/PhysRevE.93.022126

    2016 doi

  32. [40]

    2002 Entropy and temperature of a quantum Carnot engine.Proc

    Bender CM, Brody DC, Meister BK. 2002 Entropy and temperature of a quantum Carnot engine.Proc. R. Soc. Lond. A458 1519–1526. doi:10.1098/rspa.2001.0928

    2002

  33. [41]

    2011 Entropy of isolated quantum systems after a quench.Phys

    Santos LF, Polkovnikov A, Rigol M. 2011 Entropy of isolated quantum systems after a quench.Phys. Rev. Lett 107 040601. doi:10.1103/PhysRevLett.107.040601

    2011 doi

  34. [42]

    2011 Rigorous and general definition of thermodynamic entropy in thermodynamics

    Beretta GP, Zanchini E. 2011 Rigorous and general definition of thermodynamic entropy in thermodynamics. In Thermodynamics. Tadashi M, Ed., Rijeka: InTechOpen, pp. 23–50. doi:10.5772/13371

    2011 doi

  35. [43]

    2015 What is a simple system? ASME Journal of Energy Resources Technology 137 021007

    Beretta GP, Gyftopoulos EP. 2015 What is a simple system? ASME Journal of Energy Resources Technology 137 021007. doi:10.1115/1.4026383

    2015 doi

  36. [44]

    2015 Thermal equilibrium of a macroscopic quantum system in a pure state.Phys

    Goldstein S, Huse DA, Lebowitz JL, Tumulka R. 2015 Thermal equilibrium of a macroscopic quantum system in a pure state.Phys. Rev. Lett115 100402. doi:10.1103/PhysRevLett.115.100402

    2015 doi

  37. [45]

    2017 Stochastic and macroscopic thermodynamics of strongly coupled systems.Phys

    Jarzynski C. 2017 Stochastic and macroscopic thermodynamics of strongly coupled systems.Phys. Rev. X 7 011008. doi:10.1103/PhysRevX.7.011008

    2017 doi

  38. [46]

    ZanchiniE.1986Onthedefinitionofextensivepropertyenergybythefirstpostulateofthermodynamics. Found. Phys. 16 923–935. doi:10.1007/BF00765339

    doi

  39. [47]

    1988 Thermodynamics: energy of closed and open systems

    Zanchini E. 1988 Thermodynamics: energy of closed and open systems. Il Nuovo Cimento B 101 453–465. doi:10.1007/BF02828923

    1988 doi

  40. [48]

    1992 Thermodynamics: energy of nonsimple systems and second postulate

    Zanchini E. 1992 Thermodynamics: energy of nonsimple systems and second postulate. Il Nuovo Cimento B 107 123–139. doi:10.1007/BF02722911

    1992 doi

  41. [49]

    1971 General state changes in quantum theory.Ann

    Kraus K. 1971 General state changes in quantum theory.Ann. Phys. 64 311–335. doi:10.1016/ 0003- 4916(71)90108-4. doi:10.1016/0003-4916(71)90108-4

    1971 doi

  42. [50]

    1972 On quantum statistical mechanics of non-Hamiltonian systems.Rep

    Kossakowski A. 1972 On quantum statistical mechanics of non-Hamiltonian systems.Rep. Math. Phys. 3 247–274. doi:10.1016/0034-4877(72)90010-9

    1972 doi

  43. [51]

    1975 On the connection of nonequilibrium information thermodynamics with non-Hamiltonian quantum mechanics of open systems.Ann

    Ingarden RS, Kossakowski A. 1975 On the connection of nonequilibrium information thermodynamics with non-Hamiltonian quantum mechanics of open systems.Ann. Phys. 89 451–485. doi:10.1016/0003- 4916(75)90190-6

    1975 doi

  44. [52]

    1976 On the generators of quantum dynamical semigroups.Commun

    Lindblad G. 1976 On the generators of quantum dynamical semigroups.Commun. Math. Phys. 119 119–130. doi:10.1063/1.523789

    1976 doi

  45. [53]

    1976 Completely positive dynamical semigroups of N-level systems

    Gorini V, Kossakowski A, Sudarshan ECG. 1976 Completely positive dynamical semigroups of N-level systems. J. Math. Phys.17 821–825. doi:10.1063/1.522979 14

    1976 doi

  46. [54]

    SpohnH.1976ApproachtoequilibriumforcompletelypositivedynamicalsemigroupsofN-levelsystems. Rep. Math. Phys.10 189–194. doi:10.1016/0034-4877(76)90040-9

    doi

  47. [55]

    2010 Maximum entropy production rate in quantum thermodynamics.J

    Beretta GP. 2010 Maximum entropy production rate in quantum thermodynamics.J. Phys. Conf. Ser. 237 012004. doi:10.1088/1742-6596/237/1/012004

    2010 doi

  48. [56]

    2007 Well-behaved nonlinear evolution equation for steepest-entropy-ascent dissipative quantum dynamics.Int

    Beretta GP. 2007 Well-behaved nonlinear evolution equation for steepest-entropy-ascent dissipative quantum dynamics.Int. J. Quantum Inf.5 249–255. doi:10.1142/S0219749907002700

    2007 doi

  49. [57]

    2015 Steepest-entropy-ascent quantum thermodyna- mic modeling of decoherence in two different microscopic composite systems.Phys

    Cano-Andrade S, Beretta GP, von Spakovsky MR. 2015 Steepest-entropy-ascent quantum thermodyna- mic modeling of decoherence in two different microscopic composite systems.Phys. Rev. A91 013848. doi:10.1103/PhysRevA.91.013848

    2015 doi

  50. [58]

    Smith C. 2016 Comparing the models of steepest entropy ascent quantum thermodynamics, master equation and the difference equation for a simple quantum system interacting with reservoirs.Entropy 18 176. doi:10.3390/e18050176

    2016 doi

  51. [59]

    1985 Quantum thermodynamics

    Beretta GP, Gyftopoulos EP, Park JL. 1985 Quantum thermodynamics. A new equation of motion for a general quantum system.Nuovo Cimento B 87 77–97

    1985

  52. [60]

    2009 Nonlinear quantum evolution equations to model irreversible adiabatic relaxation with maximal entropy production and other nonunitary processes

    Beretta GP. 2009 Nonlinear quantum evolution equations to model irreversible adiabatic relaxation with maximal entropy production and other nonunitary processes. Rep. Math. Phys. 64 139–168. doi:10.1016/S0034-4877(09)90024-6

    2009 doi

  53. [61]

    1873 A method of geometrical representation of the thermodynamic properties by means of surfaces

    Gibbs JW. 1873 A method of geometrical representation of the thermodynamic properties by means of surfaces. Transactions of Connecticut Academy of Arts and Sciences2 382–404

  54. [62]

    1980Statistical Physics, Part I

    Landau LD, Lifshitz EM. 1980Statistical Physics, Part I. Elmsford, NY: Pergamon press

  55. [63]

    2013 Non-equilibrium thermodynamics in multiphase flows New York, NY: Springer

    Mauri R. 2013 Non-equilibrium thermodynamics in multiphase flows New York, NY: Springer. doi:10.1007/978-94-007-5461-4

    2013 doi

  56. [64]

    2015 What is the third law? ASME Journal of Energy Resources Technology 137 021004

    Beretta GP, Gyftopoulos EP. 2015 What is the third law? ASME Journal of Energy Resources Technology 137 021004. doi:10.1115/1.4026380

    2015 doi

  57. [65]

    2014 Steepest Entropy Ascent Model for Far-Non-Equilibrium Thermodynamics

    Beretta GP. 2014 Steepest Entropy Ascent Model for Far-Non-Equilibrium Thermodynamics. Uni- fied Implementation of the Maximum Entropy Production Principle. Phys. Rev. E 90 042113. doi:10.1103/PhysRevE.90.042113

    2014 doi

  58. [66]

    1971 Rate-controlled partial-equilibrium method for treating reacting gas mixtures

    Keck JC, Gillespie D. 1971 Rate-controlled partial-equilibrium method for treating reacting gas mixtures. Combustion and Flame 17 237–241. doi:10.1016/S0010-2180(71)80166-9

    1971 doi

  59. [67]

    1990 Rate-controlled constrained-equilibrium theory of chemical reactions in complex systems

    Keck JC. 1990 Rate-controlled constrained-equilibrium theory of chemical reactions in complex systems. Progress in Energy and Combustion Science16 125–154. doi:10.1016/0360-1285(90)90046-6

    1990 doi

  60. [68]

    2012 The Rate-controlled constrained-equilibrium approach to far-from-local-equilibrium thermodynamics.Entropy 14 92–130

    Beretta GP, Keck JC, Janbozorgi M, Metghalchi H. 2012 The Rate-controlled constrained-equilibrium approach to far-from-local-equilibrium thermodynamics.Entropy 14 92–130. doi:10.3390/e14020092

    2012 doi

  61. [69]

    2001 Corrections and enhancements of quasi-equilibrium states J

    Gorban AN, Karlin IV, Ilg P, Öttinger HC. 2001 Corrections and enhancements of quasi-equilibrium states J. Non-Newtonian Fluid Mech.96 203–219. doi:10.1016/S0377-0257(00)00135-X

    2001 doi

  62. [70]

    1964 On the macroscopic description of kinetic processes (in russian).Dokl

    Kogan AM, Rozonoer LI. 1964 On the macroscopic description of kinetic processes (in russian).Dokl. Akad. Nauk SSSR158 566–569

    1964

  63. [71]

    2019 Dynamic maximum entropy reductionEntropy 21 715

    Klika V, Pavelka M, Vágner P, Grmela M. 2019 Dynamic maximum entropy reductionEntropy 21 715. doi:10.3390/e21070715

    2019 doi

  64. [72]

    Li G, von Spakovsky MR. 2016 Steepest-entropy-ascent quantum thermodynamic modeling of the re- laxation process of isolated chemically reactive systems using density of states and the concept of hypoequilibrium state.Phys. Rev. E 93 012137. doi:10.1103/PhysRevE.93.012137 15

    2016 doi

  65. [73]

    2005 Maximum entropy production and the fluctuation theorem.J

    Dewar RC. 2005 Maximum entropy production and the fluctuation theorem.J. Phys. A: Math. Gen. 38 L371–L381. doi:10.1088/0305-4470/38/21/L01

    2005 doi

  66. [74]

    2013 Entropy and entropy production: Old misconceptions and new breakthroughs

    Martyushev LM. 2013 Entropy and entropy production: Old misconceptions and new breakthroughs. Entropy 15 1152–1170. doi:10.3390/e15041152

    2013 doi

  67. [75]

    1980 The Maxwell-Vlasov equations as a continuous Hamiltonian system.Phys

    Morrison PJ. 1980 The Maxwell-Vlasov equations as a continuous Hamiltonian system.Phys. Lett. A 80 383–386. doi:10.1016/0375-9601(80)90776-8

    1980 doi

  68. [76]

    1982 The Hamiltonian structure of the Maxwell-Vlasov equations.Physica D 4 394–406

    Marsden JE, Weinstein A. 1982 The Hamiltonian structure of the Maxwell-Vlasov equations.Physica D 4 394–406. doi:10.1016/0167-2789(82)90043-4

    1982 doi

  69. [77]

    1984 Dissipative hamiltonian systems: A unifying principle.Phys

    Kaufman AN. 1984 Dissipative hamiltonian systems: A unifying principle.Phys. Lett. A100, 419–422. doi:10.1016/0375-9601(84)90634-0

    1984 doi

  70. [78]

    1984 Bracket formulation for irreversible classical fields.Phys

    Morrison PJ. 1984 Bracket formulation for irreversible classical fields.Phys. Lett. A 100, 423–427. doi:10.1016/0375-9601(84)90635-2

    1984 doi

  71. [79]

    1984 Bracket formulation of dissipative fluid mechanics equations.Phys

    Grmela M. 1984 Bracket formulation of dissipative fluid mechanics equations.Phys. Lett. A102, 355–

    1984

  72. [80]

    1987 Steepest entropy ascent in quantum thermodynamics.Lecture Notes in Physics278 441–443

    Beretta GP. 1987 Steepest entropy ascent in quantum thermodynamics.Lecture Notes in Physics278 441–443. doi:10.1007/3-540-17894-5_404

    1987 doi

  73. [81]

    1985 Entropy and irreversibility for a single isolated two-level system

    Beretta GP. 1985 Entropy and irreversibility for a single isolated two-level system. New indivi- dual quantum states and new nonlinear equation of motion. Int. J. Theor. Phys. 24 119–134. doi:10.1007/BF00672647

    1985 doi

  74. [82]

    1984 A general nonlinear evolution equation for irreversible conservative approach to stable equilibrium

    Beretta GP. 1984 A general nonlinear evolution equation for irreversible conservative approach to stable equilibrium. Proceedings of the NATO Advanced Study Institute on the Frontiers of Nonequilibrium Statistical Physics, June 3-16, 1984, Santa Fe, New Mexico, Moore GT, Scull...

    1984 doi

  75. [83]

    1986 A paradigm for joined Hamiltonian and dissipative systems.Physica D 18 410–419

    Morrison PJ. 1986 A paradigm for joined Hamiltonian and dissipative systems.Physica D 18 410–419. doi:10.1016/0167-2789(86)90209-5

    1986 doi

  76. [84]

    2007 Metriplectic structure, Leibniz dynamics and dissipative systems.J

    Guha P. 2007 Metriplectic structure, Leibniz dynamics and dissipative systems.J. Math. Anal. App. 326 121–136. doi: 10.1016/j.jmaa.2006.02.023

    2007 doi

  77. [85]

    Entropy 18 304

    MaterassiM.2016Entropyasametricgeneratorofdissipationincompletemetriplecticsystems. Entropy 18 304. doi:10.3390/e18080304

    doi

  78. [86]

    1997 Dynamics and thermodynamics of complex fluids

    Grmela M, Öttinger HC. 1997 Dynamics and thermodynamics of complex fluids. I. Development of a general formalism.Phys. Rev. E 56 6620. doi:10.1103/PhysRevE.56.6620

    1997 doi

  79. [87]

    Montefusco A, Consonni, Beretta GP. 2015 Essential equivalence of the general equation for the nonequi- librium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics,Phys. Rev. E 91 042138. doi:10.1103/Ph...

    2015 doi

  80. [88]

    1998 The variational formulation of the Fokker–Planck equation

    Jordan R, Kinderlehrer D, Otto F. 1998 The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal.29, 1–17. doi:10.1137/S0036141096303359

    1998 doi

  81. [89]

    2001 The geometry of dissipative evolution equations: The porous medium equation.Commun

    Otto F. 2001 The geometry of dissipative evolution equations: The porous medium equation.Commun. Partial Differ. Equ.26, 101–174. doi:10.1081/PDE-100002243

    2001 doi

  82. [90]

    2011 A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems

    Mielke F. 2011 A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24, 1329–1346. doi:10.1088/0951-7715/24/4/016

    2011 doi

  83. [91]

    Peletier MA, Johannes Zimmer J

    Duong MH, Mark A. Peletier MA, Johannes Zimmer J. 2013 GENERIC formalism of a Vlasov- Fokker-Planck equation and connection to large-deviation principles. Nonlinearity 26 2951–2971. doi:10.1088/0951-7715/26/11/2951 16

    2013 doi

  84. [92]

    2014 On the relation between gradient flows and the large- deviation principle, with applications to Markov chains and diffusion.Potential Anal

    Mielke A, Peletier MA, Renger DRM. 2014 On the relation between gradient flows and the large- deviation principle, with applications to Markov chains and diffusion.Potential Anal. 41, 1293–1327. doi:10.1007/s11118-014-9418-5

    2014 doi

  85. [93]

    2015 Entropy production and the geometry of dissipative evolution equations.Phys

    Reina C, Zimmer J. 2015 Entropy production and the geometry of dissipative evolution equations.Phys. Rev. E 92 052117. doi:10.1103/PhysRevE.92.052117

    2015 doi

  86. [94]

    2018 Computing diffusivities from particle models out of equili- brium

    Embacher P, Dirr N, Zimmer J, Reina C. 2018 Computing diffusivities from particle models out of equili- brium. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences474(2212)

    2018

  87. [95]

    2016 Microcanonical and resource-theoretic deri- vations of the thermal state of a quantum system with noncommuting charges.Nat

    Yunger Halpern N, Faist P, Oppenheim J, Winter A. 2016 Microcanonical and resource-theoretic deri- vations of the thermal state of a quantum system with noncommuting charges.Nat. Commun. 7 12051. doi:10.1038/ncomms12051

    2016 doi

  88. [96]

    1984 Quantum thermodynamics

    Beretta GP, Gyftopoulos EP, Park JL, Hatsopoulos GN. 1984 Quantum thermodynamics. A new equation of motion for a single constituent of matter. Nuovo Cimento B 82 169–191. doi:10.1007/BF02732871

    1984 doi

  89. [97]

    2001 Nonlinear quantum evolution with maximal entropy productionPhys

    Gheorghiu-Svirschevski S. 2001 Nonlinear quantum evolution with maximal entropy productionPhys. Rev. A 63 022105. doi:10.1103/PhysRevA.63.022105

    2001 doi

  90. [98]

    Nonlinear quantum evolution with maximal entropy production

    Gheorghiu-Svirschevski S. 2001 Addendum to “Nonlinear quantum evolution with maximal entropy production.”Phys. Rev. A63 054102. doi:10.1103/PhysRevA.63.054102

    2001 doi

  91. [99]

    1996 Microscale theory of surface tension

    Antanovskii LK. 1996 Microscale theory of surface tension. Phys. Rev. E 54 6285. doi:10.1103/PhysRevE.54.6285

    1996 doi

  92. [100]

    2001 Weakly nonlocal irreversible thermodynamics—the Guyer–Krumhansl and the Cahn–Hilliard equations.Phys

    Ván P. 2001 Weakly nonlocal irreversible thermodynamics—the Guyer–Krumhansl and the Cahn–Hilliard equations.Phys. Lett. A290 88–92. doi:10.1016/S0375-9601(01)00657-0

    2001 doi

  93. [101]

    2008 Internal Variables and Dynamic Degrees of Freedom.J

    Ván P, Berezovski A, Engelbrecht J. 2008 Internal Variables and Dynamic Degrees of Freedom.J. Non-Eq. Thermodynamics 33 235–254. doi:10.1515/JNETDY.2008.010

    2008 doi

  94. [102]

    1987 From a least action principle to mass action law and extended affinity.Chem

    Sieniutycz S. 1987 From a least action principle to mass action law and extended affinity.Chem. Eng. Sci. 42, 2697–2711. doi:10.1016/0009-2509(87)87020-3

    1987 doi

  95. [103]

    1986 A theorem on Lyapunov stability for dynamical systems and a conjecture on a property of entropyJ

    Beretta GP. 1986 A theorem on Lyapunov stability for dynamical systems and a conjecture on a property of entropyJ. Math. Phys.27 305–308. doi:10.1063/1.527390

    1986 doi

  96. [104]

    2001 Change, time and information geometry

    Caticha A. 2001 Change, time and information geometry. AIP Conf. Proc. 568 72. doi:10.1063/1.1381872

    2001 doi

  97. [105]

    2005 Nonlinear extensions of Schroedinger-von Neumann quantum dynamics: A set of necessary conditions for compatibility with thermodynamics

    Beretta GP. 2005 Nonlinear extensions of Schroedinger-von Neumann quantum dynamics: A set of necessary conditions for compatibility with thermodynamics. Mod. Phys. Lett. A 20 977–984. doi:10.1142/S0217732305017263

    2005 doi

  98. [106]

    2019 Time–energy and time–entropy uncertainty relations in nonequilibrium quan- tum thermodynamics under steepest-entropy-ascent nonlinear master equations

    Beretta GP. 2019 Time–energy and time–entropy uncertainty relations in nonequilibrium quan- tum thermodynamics under steepest-entropy-ascent nonlinear master equations. Entropy 21 679. doi:10.3390/e21070679

    2019 doi

  99. [107]

    1987 Quantum thermodynamics of nonequilibrium

    Beretta GP. 1987 Quantum thermodynamics of nonequilibrium. Onsager reciprocity and dispersion- dissipation relations.Found. Phys. 17 365–381. doi:10.1007/BF00733374

    1987 doi

  100. [108]

    1985 Effect of irreversible atomic relaxation on resonance fluorescence, absorption and stimulated emission.Int

    Beretta GP. 1985 Effect of irreversible atomic relaxation on resonance fluorescence, absorption and stimulated emission.Int. J. Theor. Phys.24 1233–1258. doi:10.1007/BF00670336

    1985 doi

  101. [109]

    2006 Nonlinear model dynamics for closed-system, constrained, maximal- entropy-generation relaxation by energy redistribution

    Beretta GP. 2006 Nonlinear model dynamics for closed-system, constrained, maximal- entropy-generation relaxation by energy redistribution. Phys. Rev. E 73 026113. doi:10.1103/PhysRevE.73.026113 17

    2006 doi

  102. [110]

    2017 Model dynamics for quantum computing

    Tabakin F. 2017 Model dynamics for quantum computing. Ann. Phys. 383 33–78. doi:10.1016/j.aop.2017.04.013

    2017 doi

  103. [111]

    BhattacharyaS,MisraA,MukhopadhyayC,PatiAK.2017Exactmasterequationforaspininteracting with a spin bath: Non-Markovianity and negative entropy production rate.Phys. Rev. A 95 012122. doi:10.1103/PhysRevA.95.012122

    doi

  104. [112]

    2010 Entropy production as correlation between system and reservoir.New J

    Esposito M, Lindenberg K, Van den Broeck C. 2010 Entropy production as correlation between system and reservoir.New J. Phys.12 013013. doi:10.1088/1367-2630/12/1/013013

    2010 doi

  105. [113]

    1985 Uniting mechanics and statistics.Nature 316 11

    Maddox J. 1985 Uniting mechanics and statistics.Nature 316 11. doi:10.1038/316011a0 18

    1985 doi

  106. [358]

    doi:10.1016/0375-9601(84)90297-4

    doi

  107. [694]

    doi:10.1098/rspa.2017.0694

    2017

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