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REVIEW 3 major objections 5 minor 266 references

Nonequilibrium and Irreversibility

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper argues that a Chaotic Hypothesis — treating chaotic nonequilibrium evolution as Anosov on its attracting set — extends ensemble theory to stationary nonequilibrium and yields universal fluctuation relations such as the…

desk verdict A mature, honest restatement of the Chaotic Hypothesis program; the load-bearing conjecture remains unproven and its scope is left vague, but the book deserves serious engagement. read the letter →

arxiv 2501.12426 v1 pith:VTD7POJC submitted 2025-01-21 nlin.CD

Giovanni Gallavotti This is my paper · ORCID
classification nlin.CD MSC 37D2037D4582C0382C05
keywords nonequilibriumstatisticalmechanicschaotichypothesisSRBdistributionAnosovsystemsfluctuationtheoremtimereversalentropyproductionphasespacecontraction
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 works toward a general, model-independent theory of stationary nonequilibrium states, modeled on equilibrium thermodynamics. Its central proposal is the 'Chaotic Hypothesis': for empirically chaotic evolution, the dynamics restricted to a transitive attracting set can be treated as an Anosov system for the purpose of computing statistical properties. If this hypothesis is accepted, the SRB distribution becomes the unique, protocol-independent distribution of a stationary state, playing the role that microcanonical and canonical ensembles play at equilibrium. Time-reversal symmetry then makes irreversibility a statistical consequence of chaotic motion itself, not of viscous forces, and yields general fluctuation relations for entropy production. The book applies the framework to thermostatted fluids, granular materials, and turbulence, and considers cases where the hypothesis might fail.

What carries the argument

The load-bearing machinery is the Chaotic Hypothesis paired with the SRB theory of uniformly hyperbolic systems. An Anosov system is a hyperbolic map or flow with a dense orbit and a dense set of periodic points; on such systems the SRB distribution is the invariant measure that gives the time averages of observables for all chaotic initial data in an attraction domain. The associated phase-space contraction $\sigma(x)$, defined up to a time derivative, has time average $\sigma_+$ independent of the metric; its positivity is the signature of genuine nonequilibrium. Markov partitions convert the dynamics into a symbolic system, coarse-graining phase space into rectangles whose SRB weights are counted by recurrent microcells, which is how the book connects the abstract invariant measure to simulations and discrete phase space.

What would settle it

A direct falsifier is a careful numerical experiment on a stationary thermostatted nonequilibrium system: compute the probability $\pi_\tau(p)$ of the normalized finite-time entropy production $p$ over long windows $\tau$ and test whether $\log[\pi_\tau(p)/\pi_\tau(-p)]/(\tau p)$ approaches the mean phase-space contraction $\sigma_+$. A persistent, systematic departure from this straight-line relation at long times, in a system whose motion is still chaotic, would show that the Chaotic Hypothesis is not valid for that system.

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Extended reading notes

Core claim

On the book's own terms, the central discovery is that every stationary state of a chaotic system is governed by a unique SRB distribution on an attracting set, provided the Chaotic Hypothesis holds. In equilibrium this distribution reduces to the ordinary microcanonical distribution, so the new principle extends rather than contradicts ensemble theory. Away from equilibrium, the average phase-space contraction rate $\sigma_+$ equals the entropy production rate, and time-reversal symmetry forces a large-deviation symmetry: the probability $\pi_\tau(p)$ of observing a time-averaged normalized entropy production $p$ over a long window $\tau$ obeys $\pi_\tau(p)/\pi_\tau(-p)=\exp(\tau\,\sigma_+ p)$. From this fluctuation relation the book derives Onsager reciprocity and Green–Kubo formulas, so linear response appears as a corollary of chaotic dynamics rather than a separate transport assumption. Irreversibility is therefore traced to the chaotic dispersal of phase-space volume under reversible equations of motion.

Load-bearing premise

The load-bearing premise is that an empirically chaotic nonequilibrium system can be modeled as uniformly hyperbolic on its attracting set, so that a unique SRB distribution describes its stationary statistics.

Editorial extensions

If this is right

  • If the Chaotic Hypothesis holds, stationary nonequilibrium statistics are independent of the preparation protocol: any smooth initial density on an attraction domain converges to the same SRB distribution.
  • Time reversal plus the Chaotic Hypothesis gives the fluctuation theorem as a universal relation, so large entropy-production fluctuations obey a known symmetry arbitrarily far from equilibrium.
  • Onsager reciprocity and the Green–Kubo formula follow from the same SRB structure, placing linear response on a dynamical foundation rather than a separate transport postulate.
  • The average phase-space contraction $\sigma_+$ measures the entropy production rate, so positive average contraction identifies a nonequilibrium state and zero contraction identifies an equilibrium-like state.
  • The framework yields quantitative criteria for quasi-staticity and irreversibility in concrete systems, from thermostatted fluids to granular materials.

Reading between the lines

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

  • Beyond the book's claims, the thermostat-equivalence theorem suggests a stronger, untested conjecture: in the thermodynamic limit the stationary SRB distributions of finite Gaussian thermostats should converge to those of infinite Newtonian thermostats, not just their finite-time trajectories.
  • Beyond the book's claims, the fluctuation theorem's observable symmetry window could serve as a practical diagnostic of effective hyperbolicity, letting simulations measure how close a real system is to the Anosov idealization.
  • Beyond the book's claims, if SRB distributions are singular attractors, adding small noise should produce a nearby smooth measure with slightly modified averages; comparing noisy and noiseless stationary states may delimit where the Chaotic Hypothesis is quantitatively reliable.
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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

3 major / 5 minor

Summary. This is the second, revised edition of a monograph by G. Gallavotti arguing that the Sinai-Ruelle-Bowen (SRB) theory of uniformly hyperbolic dynamical systems supplies the natural generalization of equilibrium ensemble theory to stationary nonequilibrium states. The load-bearing premise is the Chaotic Hypothesis (CH) of Section 2.7: empirically chaotic evolutions are hyperbolic, and on each of the finitely many transitive attracting sets the dynamics may be regarded as Anosov for statistical purposes. Under CH, the SRB distribution is the unique, protocol-independent stationary statistics; the phase-space contraction rate is identified, up to time derivatives, with the entropy production rate (Section 2.8); and the general relations of Chapter 4 follow: the Gallavotti-Cohen fluctuation theorem, fluctuation patterns, Onsager reciprocity, and Green-Kubo relations. Chapter 1 and Chapter 6 give the historical development, including a modern proof of the heat theorem in Section 1.4 and translations of Boltzmann and Clausius; Chapter 3 develops the discrete symbolic-dynamics picture of SRB statistics; and Chapter 5 applies the framework conjecturally to thermostats, irreversibility measures, fluids, turbulence, stochastic evolutions, and granular matter. The book is deliberately informal and heuristic, with the rigorous anchors (SRB theorem, Anosov/Axiom A results) cited from the literature.

Significance. The conditional claim at the heart of the book is defensible and, on the book's own terms, well executed: if a system satisfies CH, the SRB machinery yields unique stationary statistics, and the Gallavotti-Cohen fluctuation theorem is mathematically sound for Anosov systems, as the cited literature confirms. Concrete strengths include the thermostat-equivalence estimate of Eq. (2.5.2), the exact stationary distributions for single free Gaussian thermostats in Section 2.8(e), the parameter-free heat-theorem derivation of Section 1.4, and the translations of Boltzmann and Clausius in Chapter 6. The book is unusually honest about its epistemic status: CH is labeled a hypothesis with conceded counterexamples (Section 2.7), the initial-data hypothesis and Axiom C are stated as separate assumptions, and Chapter 5 is explicitly conjectural. The stress-test concern does land on reading the manuscript: no criterion delimits the empirically chaotic systems to which CH is asserted to apply, so the abstract's 'general and model independent' wording overstates the domain even relative to the book's own caveats, and the fluid and granular applications add further unverified layers.

major comments (3)
  1. [Abstract; §2.7; §2.6] The stress-test concern about the unspecified domain of CH lands on reading the manuscript. The abstract's claim of 'general and model independent' relations is not supported by the book's own scope statements: Section 2.7 asserts CH only for 'empirically chaotic evolutions,' immediately concedes that 'simple counterexamples exist,' and Section 2.6 restricts the entire treatment to hyperbolic systems, explicitly setting aside attracting surfaces that are not hyperbolic ('However here we shall only consider hyperbolic systems'). Because Section 5.6 (intermittency, phase transitions) acknowledges mechanisms outside the finite-union-of-Anosov-attractors picture, and Section 2.7 remark (5) treats attracting periodic orbits as excludable only 'for simplicity,' the class of systems to which the SRB-based predictions are meant to apply is never delimited. The manuscript should either state a domain criterion — for example, an explicit enumeration of the counterexample classes (indifferent fixed points, neutral directions, intermittent maps) that are presumed absent, with physical reasons — or systematically rephrase the universality claims as conditional on CH, e.g., 'for systems satisfying the Chaotic Hypothesis.' This is load-bearing because the fluctuation theorem, fluctuation patterns, and Onsager relations of Chapter 4 inherit their claimed generality entirely from CH.
  2. [§5.4–§5.6; Appendices J–L] The fluid and granular applications require additional assumptions beyond CH as stated, and these assumptions should be identified as separate conjectures. CH is formulated for finite-dimensional smooth phase spaces (Section 2.6, Definition 0), whereas the Navier-Stokes treatment in Sections 5.4–5.5 and Appendices J–L passes to regularized equations and conjectures that properties 'might survive the regularization removal' (Preface): this is a regularization-independence conjecture for SRB-based statistics that is nowhere stated as a distinct hypothesis. Similarly, Sections 5.10–5.12 presuppose finitely many transitive attracting sets for dissipative granular dynamics, a condition that is itself part of CH and is unverified for those models. The chapter title 'Conjectures and suggested applications' labels this material appropriately, but the abstract's mention of the incompressible fluid as a case treated within the same theory obscures the extra layer of assumption. By contrast, the thermostat-equivalence gap in Section 2.5 is handled honestly: the short-time theorem (2.5.2) is stated as a theorem, and the equality of stationary distributions in the thermodynamic limit is explicitly conceded as unproved. The fluid and granular extensions deserve the same explicitness: each should state which claims are consequences of CH, which require additional conjectures, and which are expected to survive in modified form.
  3. [§3.7–§3.8] The microcell derivation of the SRB weights in Section 3.8 is presented as determining the physical meaning of the SRB distribution, but it rests on discretization postulates (a)–(d) of Section 3.7 that the text itself flags as 'not innocent': the existence of a single-cycle permutation for a 'careful enough' approximating map, the placement of recurrent microcells on finitely many unstable axes per coarse cell, and uniform microcell spacing. These postulates are not implied by CH. For genuine Anosov systems, the weight formula w(q) proportional to the unstable expansion factor is a rigorous consequence of the Gibbs property of the SRB measure — with absolutely continuous conditional measures on unstable manifolds — so the book should separate those rigorous ingredients from the heuristic discretization and indicate which parts of the computation would survive a change of the approximating program. As written, a reader cannot tell whether the microcell computation is a diagram of the SRB theorem or an independent conjecture about digital representations of Anosov systems, and the phrase 'determines an approximation of the weights' at the end of Section 3.7 invites a reading that the text then does not justify.
minor comments (5)
  1. [§1.4] The cross-reference 'Sec.(refsec:V-6)' in Section 1.4 is a broken LaTeX reference and should be replaced by the actual section number in the final version.
  2. [Preface and Table of Contents] The Preface refers to appendices Q–T as reporting work in progress on the BBGKY hierarchy, but the Table of Contents assigns the labels Q and R to the Citations index and the Chapter abstracts; the appendix numbering should be reconciled between the Preface and the front matter.
  3. [Title page; Abstract; §1.8; §2.7] There are several typographical errors to correct: 'Chaothic' for 'Chaotic' in the Abstract, 'Spriner-Nature' for 'Springer-Nature' on the title page, 'apported' for 'made' in the Preface, 'fluctation theorem' for 'fluctuation theorem' in the Table of Contents, 'theoren' for 'theorem' in Section 1.8, and 'misundertandings' for 'misunderstandings' in Section 2.7.
  4. [§2.6 running head] The running head of Section 2.6 ('SRB, attracting surfaces, Anosov's evolution and chaotic data') differs from the section title in the body and in the Table of Contents ('Hyperbolicity, attracting surfaces, Anosov maps, chaotic data'); the three should be made consistent.
  5. [§3.5] In Section 3.5, 'its eigenvalues equation is' should read 'its eigenvalue equation is'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the book's derivations are conditional on the explicitly stated Chaotic Hypothesis, with the SRB machinery and fluctuation theorem resting on externally grounded mathematical results rather than on fitted or self-referential inputs.

full rationale

The central load-bearing premise is the Chaotic Hypothesis (CH), stated in Section 2.7 as an explicit assumption about empirically chaotic evolutions, not as a derived consequence of the fluctuation theorem or of anything else in the book. The text acknowledges that 'simple counterexamples exist' and repeatedly treats CH as a hypothesis to be used heuristically, so the derivation of SRB-based predictions from CH is a conditional derivation from a stated premise, not a circular definition. The SRB theorem invoked to select the stationary distribution is attributed to Sinai, Ruelle, and Bowen, i.e., to external mathematical work, and the book uses it as a tool rather than importing a uniqueness claim from the author's own prior papers. The Gallavotti-Cohen fluctuation theorem is presented as a consequence of CH plus time-reversal symmetry, not as a renaming of the hypothesis or as a fitted parameter renamed as a prediction. Self-citations appear in naming and historically attributing CH and the fluctuation theorem, but the book does not use those self-citations to justify the central claim; it openly labels CH as a conjecture. Where the book notes limitations, such as non-hyperbolic attracting surfaces or possible violations of CH, these affect the physical validity or scope of the conclusions, but they do not indicate that any equation reduces to its own input by construction. There is no exhibited step in which an output quantity is defined in terms of the quantity it is supposed to predict, no fitted parameter is relabeled as a prediction, and no external theorem is replaced by a self-citation chain. The main scientific risk is the unresolved status and scope of CH itself, which is a correctness concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The book's conclusions rest on the Chaotic Hypothesis, which is an unproven conjecture, plus standard hyperbolicity theorems. No free parameters are fitted to data, and no new physical entities are invented.

assumptions (4)
  • ad hoc to paper Chaotic Hypothesis: dynamics on phase space is hyperbolic, and on a transitive attracting set it can be regarded as an Anosov system for statistical purposes.
    Section 2.7 introduces CH as the foundational assumption of the book. It is a conjecture proposed by the author and Cohen, not a proven property of physical systems.
  • domain assumption Initial data have a continuous density on the constraint surface.
    Section 2.4 formalizes the initial data hypothesis: physical preparation produces initial distributions with density. This is assumed to make the SRB limit independent of the protocol.
  • ad hoc to paper Equivalence of finite Gaussian thermostats and infinite Newtonian thermostats in the thermodynamic limit.
    Section 2.5 proves only finite-time closeness (Eq. 2.5.2) and states the stationary-state equivalence in the thermodynamic limit as an expectation, not a theorem.
  • ad hoc to paper Axiom C: symmetry of SRB distributions under time reversal and other operations needed for Onsager reciprocity.
    Appendix H and Section 4.3 introduce Axiom C as an additional assumption beyond CH to derive fluctuation patterns and Onsager reciprocity. The excerpt mentions it without proof.

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Pith. "Pith review of Nonequilibrium and Irreversibility." pith.science (2026). https://pith.science/paper/VTD7POJC

@misc{pith2026250112426,
  author       = {Pith},
  title        = {Pith review of: Nonequilibrium and Irreversibility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTD7POJC}},
  note         = {Machine review of arXiv:2501.12426}
}
read the original abstract

The work concentrates on relations, which are general and model independent in chaotic system, between time averages of a few (typically {\it very few}) observables. Equilibrium thermodynamics provides a guide and here is attempted to argue that the viewpoint of Sinai-Ruelle-Bowen can be regarded as a generalization to nonequilibrum phenomena of the theory of the ensembles proposing an answer to classical question like which distributions describe the statistics of stationary states (hence extend the analysis selecting canonical, or equivalent distributions, equilibrim between the uncountably many possibilities). The special name "Chaothic Hypothesis" (CH) is given to the above attempt and its mathematical meaning is discussed. General properties are presented and applied (eg. 'Fluctuation Theorem', 'Fluctuation Patterns', 'Pairing Symmetry') and related to the basic Time Reversal symmetry: which presents irreversibility as due to chaotic motion rather than to viscous forces. The case of a simple incompressible fluid is discussed in some detail. The possibility that CH is violated in various cases is considered: and in the end it is suggested that CH is the paradigm of chaotic evolution, as the harmonic oscillators are a paradigm of ordered motions, but of course {\it tertium datur}. The exposition is informal and often restricted to heuristic analysis, with detailed references to the literature and attention to numerical simulations and importance of stressing strongly the discrete models of Physics, trying to imitate the vision of Boltzmann, is widely considered.

Figures

Figures reproduced from arXiv: 2501.12426 by the authors.

Figure 2.2
Figure 2.2. Fig.2.2.1: Finite thermostats model (Gaussian thermostat [PITH_FULL_IMAGE:figures/full_fig_p037_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Fig.2.3.1: A model for electric conduction. The container [PITH_FULL_IMAGE:figures/full_fig_p040_2_3.png] view at source ↗
Figure 2.2
Figure 2.2. Fig.2.2.1 with [PITH_FULL_IMAGE:figures/full_fig_p054_2_2.png] view at source ↗
Figures from the paper (14 more)
Figure 3.3
Figure 3.3. Figure 3.3: fig3.3.1 [PITH_FULL_IMAGE:figures/full_fig_p066_3_3.png]
Figure 3.3
Figure 3.3. Figure 3.3: fig3.3.3 [PITH_FULL_IMAGE:figures/full_fig_p067_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: Fig.3.4.1: The figures illustrate very symbolically, as 2- [PITH_FULL_IMAGE:figures/full_fig_p069_3_4.png]
Figure 3.4
Figure 3.4. Figure 3.4: Fig.3.4.1 above): imagine each [PITH_FULL_IMAGE:figures/full_fig_p070_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: fig3.5.1 [PITH_FULL_IMAGE:figures/full_fig_p072_3_5.png]
Figure 3.7
Figure 3.7. Figure 3.7: Fig3.7.1 [PITH_FULL_IMAGE:figures/full_fig_p076_3_7.png]
Figure 3.10
Figure 3.10. Figure 3.10: Fig.3.10.1: the shadowed region represents the intersecti [PITH_FULL_IMAGE:figures/full_fig_p083_3_10.png]
Figure 5.2
Figure 5.2. Figure 5.2: Fig.5.2.1 [PITH_FULL_IMAGE:figures/full_fig_p118_5_2.png]
Figure 5.2
Figure 5.2. Figure 5.2: Fig.5.2.3: The wavy lines symbolizes the surface of the wate [PITH_FULL_IMAGE:figures/full_fig_p119_5_2.png]
Figure 5.2
Figure 5.2. Figure 5.2: Fig.5.2.4 [PITH_FULL_IMAGE:figures/full_fig_p121_5_2.png]
Figure 5.2
Figure 5.2. Figure 5.2: Fig.5.2.4: An example of a process proceeding at jumps of siz [PITH_FULL_IMAGE:figures/full_fig_p122_5_2.png]
Figure 5.9
Figure 5.9. Figure 5.9: Fig.5.9.1: The (microscopic) sensor is attached to the arm [PITH_FULL_IMAGE:figures/full_fig_p141_5_9.png]
Figure 6.1
Figure 6.1. Figure 6.1: Fig.6.1.2: spherical triangle for momentum conservation. [PITH_FULL_IMAGE:figures/full_fig_p211_6_1.png]
Figure 3
Figure 3. Figure 3: Fig.3.A.1: An incomplete rectangle [PITH_FULL_IMAGE:figures/full_fig_p225_3.png]

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Works this paper leans on

266 extracted references · 79 canonical work pages

  1. [1]

    Boltzmann

    L. Boltzmann. Wissenschaftliche Abhandlungen, ed. F. Hasen¨ ohrl , volume 1,2,3 (http://www.esi.ac.at/further/Boltzmann online.html). Barth, Leipzig, 1909

    1909

  2. [2]

    Dictionnaire Philosophique

    G. Gallavotti. L’ hypoth` ese ergodique et Boltzmann. In “Dictionnaire Philosophique” (ed. K. Chemla), Presses Universitaires, P aris, pages 1081– 1086, 1989

    1989

  3. [3]

    Gallavotti

    G. Gallavotti. Trattatello di Meccanica Statistica , volume 50. Quaderni del CNR-GNFM, Firenze, 1995

    1995

  4. [4]

    Gallavotti

    G. Gallavotti. Ergodicity, ensembles, irreversibility in Boltzmann and beyond. Journal of Statistical Physics , 78:1571–1589, 1995

    1995

  5. [5]

    Gallavotti

    G. Gallavotti. Statistical Mechanics. A short treatise . Springer Verlag, Berlin, 2000

    2000

  6. [6]

    Gallavotti

    G. Gallavotti. Equilibrium statistical mechanics. Encyclopedia of Mathematical Physics, ed. J.P. Fran¸ coise, G.L. Naber, Tsou Sheung Tsun , 1:51–87, 2006

    2006

  7. [7]

    Bertini, A

    L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim. Fluc- tuations in stationary nonequilibrium states of irreversi ble processes. Physical Review Letters, 87:040601, 2001

    2001

  8. [8]

    Derrida, J

    B. Derrida, J. L. Lebowitz, and E. R. Speer. Exact free ener gy functional for a driven diffusive open stationary nonequilibrium syste m. Physical Review Letters, 89:030601, 2002

    2002

Show all 266 references
  1. [9]

    Gerschenfeld, B

    A. Gerschenfeld, B. Derrida, and J. L. Lebowitz. Anomalou s Fourier’s law and long range correlations in a 1D non-momentum conserving mec hanical model. Journal of Statistical Physics , 141:757–766, 2010

    2010

  2. [10]

    Bricmont and A

    J. Bricmont and A. Kupiainen. Diffusion in energy conserv ing coupled maps. Communications in Mathematical Physics , 321:311–369, 2013

    2013

  3. [11]

    Feynman and F.L

    R.P. Feynman and F.L. Vernon. The theory of a general quan tum system interacting with a linear dissipative system. Annals of Physics , 24:118–173, 1963

    1963

  4. [12]

    Bonetto, G

    F. Bonetto, G. Gallavotti, A. Giuliani, and F. Zamponi. C haotic Hypothesis, Fluctuation Theorem and Singularities. Journal of Statistical Physics , 123:39– 54, 2006

    2006

  5. [13]

    E. Lorenz. Deterministic non periodic flow. Journal of the Atmospheric Sci- ence, 20:130–141, 1963

    1963

  6. [14]

    Lucretius

    T. Lucretius. De Rerum Natura . Rizzoli, Milano, 1976. References 234

    1976

  7. [15]

    S.G. Brush. History of modern physical sciences, Vol.I: The kinetic the ory of gases. Imperial College Press, London, 2003

    2003

  8. [16]

    I. Newton. Philosophiae Naturalis Principia Methematica . Royal Society, Am- sterdam, 1723

  9. [17]

    P.S. Laplace. Sur l’attraction des corps sph´ eriques etsur la r´ epulsion des fluides ´ elastiques.Annales de Chimie , 18:181–189, 1821

  10. [18]

    Avogadro

    A. Avogadro. Essai d’une mani` ere de determiner les mass es relatives des molecules ´ el´ ementaires des corps, et les proportions sel on lesquelles elles en- trent dans ces combinaisons. Journal de Physique, de Chimie et d’Histoire naturelle, translated in lem.ch.unito.it/ch...

  11. [19]

    S.G. Brush. The kind of motion that we call heat, (I, II) . North Holland, Amsterdam, 1976

    1976

  12. [20]

    Mossotti

    O.F. Mossotti. Sur les forces qui r´ egissent la constitution int´ erieure des corps. Stamperia Reale, Torino, 1836

  13. [21]

    Clausius

    R. Clausius. On the motive power of heat, and on the laws wh ich can be deduced from it for the theory of heat. Philosophical Magazine, 2:1–102, 1851

  14. [22]

    S. Carnot. R´ eflections sur la puissance motrice du feu et sur les machin es propres ` a d´ evelopper cette puissance. Online in https://gallica.bnf.fr; original Bachelier, 1824; reprinted Gabay, 1990., Paris, 1824

    1990

  15. [23]

    Clausius

    R. Clausius. ¨Uber einige f¨ ur Anwendung bequeme formen der Hauptgle- ichungen der mechanischen W¨ armetheorie. Annalen der Physik und Chemie , 125:353–401, 1865

  16. [24]

    Lidddell and R

    H.G. Lidddell and R. Scott. A Greek-English Lexicon . Oxford, Oxford, 1968

    1968

  17. [25]

    Kr¨ onig

    A. Kr¨ onig. Grundz¨ uge einer Theorie der Gase. Annalen der Physik und Chemie, XCIX:315–322, 1856

  18. [26]

    Clausius

    R. Clausius. The nature of the motion which we call heat. Philosophical Magazine, 14:108–127, 1865

  19. [27]

    J.C. Maxwell. Illustrations of the dynamical theory of g ases. In: The Scientific Papers of J.C. Maxwell, Cambridge University Press, Ed. W.D . Niven, Vol.1 , pages 377–409, 1860

  20. [28]

    J.C. Maxwell. The Scientific Papers of J.C. Maxwell . Cambridge University Press, Ed. W.D. Niven, Vol.1,2, Cambridge, 1964

    1964

  21. [29]

    Boltzmann

    L. Boltzmann. ¨Uber die mechanische Bedeutung des zweiten Hauptsatzes der W¨ armetheorie.Wiener Berichte , 53, (W.A.,#2):195–220, (9–33), 1866

  22. [30]

    Boltzmann

    L. Boltzmann. Theoretical Physics and philosophical writings, ed. B. Mc G uin- ness. Reidel, Dordrecht, 1974

    1974

  23. [31]

    J. C. Maxwell. On Boltzmann’s theorem on the average dist ribution of energy in a system of material points. Transactions of the Cambridge Philosophical Society, 12:547–575, 1879

  24. [32]

    Boltzmann

    L. Boltzmann. Weitere Studien ¨ uber das W¨ armegleichge wicht unter Gas- molek¨ ulen.Wiener Berichte , 63, (W.A.,#22):275–370, (316–402), 1872

  25. [33]

    J.C. Maxwell. On the dynamical theory of gases. In: The Scientific Papers of J.C. Maxwell, Cambridge University Press, Ed. W.D. Niven , Vol.2 , pages 26–78, 1866

  26. [34]

    Clausius

    R. Clausius. Ueber die Zur¨ uckf¨ uhrung des zweites Haup tsatzes der mecha- nischen W¨ armetheorie und allgemeine mechanische Prinzip ien. Annalen der Physik, 142:433–461, 1871

  27. [35]

    Boltzmann

    L. Boltzmann. Zur priorit¨ at der auffindung der beziehung zwischen dem zweiten hauptsatze der mechanischen w¨ armetheorie und dem prinzip der kle- insten wirkung. Poggendorf Annalen , 143, (W.A.,#17):211–230, (228–236), 1871. References 235

  28. [36]

    Clausius

    R. Clausius. Bemerkungen zu der priorit¨ at reclamation des hrn. boltzmann. Annalen der Physik , 144:265–280, 1872

  29. [37]

    Boltzmann

    L. Boltzmann. ¨Uber das W¨ armegleichgewicht zwischen mehratomigen Gas- molek¨ ulen.Wiener Berichte , 68, (W.A.,#18:397–418, (237–258), 1871

  30. [38]

    Boltzmann

    L. Boltzmann. Einige allgemeine s¨ atze ¨ uber W¨ armegle ichgewicht. Wiener Berichte, 63, (W.A.,#19):679–711, (259–287), 1871

  31. [39]

    Boltzmann

    L. Boltzmann. Analytischer Beweis des zweiten Hauptsat zes der mechanischen W¨ armetheorie aus den S¨ atzen ¨ uber das Gleichgewicht des lebendigen Kraft. Wiener Berichte , 63, (W.A.,#20):712–732,(288–308), 1871

  32. [40]

    Boltzmann

    L. Boltzmann. ¨Uber die Beziehung zwischen dem zweiten Hauptsatze der mech - anischen W¨ armetheorie und der Wahrscheinlichkeitsrechnung, respektive den S¨ atzen ¨ uber das W¨ armegleichgewicht.Wiener Berichte , 76, (W.A.,#42):373– 435, (164–223), 1877

  33. [41]

    Boltzmann

    L. Boltzmann. ¨Uber die Eigenshaften monozyklischer und anderer damit ver - wandter Systeme. Crelles Journal , 98, (W.A.,#73):68–94, (122–152), 1884

  34. [42]

    Lagrange

    J.L. Lagrange. Oeuvres. Gauthiers-Villars, Paris, 1867-1892

  35. [43]

    Boltzmann

    L. Boltzmann. Studien ¨ uber das gleichgewicht der leben digen kraft zwischen bewegten materiellen punkten. Wiener Berichte , 58, (W.A.,#5):517–560, (49– 96), 1868

  36. [44]

    Boltzmann

    L. Boltzmann. Bemerkungen ¨ uber einige probleme der mec hanischen W¨ arme- theorie. Wiener Berichte , 75, (W.A.,#39):62–100, (112–148), 1877

  37. [45]

    Gallavotti

    G. Gallavotti. Quasi periodic motions from Hypparchus t o Kolmogorov. Ren- diconti Accademia dei Lincei, Matematica e applicazioni , 12:125–152, 2001

    2001

  38. [46]

    A. Bach. Boltzmann’s probability distribution of 1877. Archive for the History of exact Sciences , 41:1–40, 1990

    1990

  39. [47]

    Boltzmann

    L. Boltzmann. L¨ osung eines mechanischen problems. Wiener Berichte , 58, (W.A.,#6):1035–1044, (97–105), 1868

  40. [48]

    J. Gibbs. Elementary principles in statistical mechanics . Schribner, Cambridge, 1902

    1902

  41. [49]

    Boltzmann

    L. Boltzmann. Zusammenhang zwischen den S¨ atzen ¨ uber d as Verhalten mehratomiger Gasmolek¨ ule mit Jacobi’s Prinzip des letzte n Multiplicators. Wiener Berichte , 63, (W.A.,#19):679–711, (259–287), 1871

  42. [50]

    Gallavotti, F

    G. Gallavotti, F. Bonetto, and G. Gentile. Aspects of the ergodic, qualitative and statistical theory of motion . Springer Verlag, Berlin, 2004

    2004

  43. [51]

    O. Lanford. Time evolution of large classical systems. Dynamical systems, theory and applications, Lecture Notes in Physics, ed. J. Mo ser, 38:1–111, 1974

    1974

  44. [52]

    H. Spohn. On the integrated form of the BBGKY hierarchy fo r hard spheres. arxiv: math-ph/0605068 , pages 1–19, 2006

    2006 arXiv

  45. [53]

    W. Thomson. The kinetic theory of dissipation of energy. Proceedings of the Royal Society of Edinburgh , 8:325–328, 1874

  46. [54]

    J.W. Shim. A commented translation of Boltzmann’s work, ueber die sogenan- nte h-curve . European Physics Journal H , 49:21, 2024

    2024

  47. [55]

    Boltzmann

    L. Boltzmann. Ueber die sogenannte h-curve. Mathematische Annalen, and W.A. #128 , 50:325–332, 1898

  48. [56]

    Boltzmann

    L. Boltzmann. Lectures on gas theory, English edition annotated by S. Brus h. University of California Press, Berkeley, 1964

    1964

  49. [57]

    Goldstein and J

    S. Goldstein and J. L. Lebowitz. On the (Boltzmann) entro py of nonequilibrium systems. Physica D , 193:53–66, 2004

    2004

  50. [58]

    P. L. Garrido, S. Goldstein, and J. L. Lebowitz. Boltzman n entropy for dense fluids not in local equilibrium. Physical Review Letters , 92:050602 (+4), 2004. References 236

    2004

  51. [59]

    Gallavotti

    G. Gallavotti. Counting phase space cells in statistica l mechanics. Communi- cation in Mathematical Physics , 224:107–112, 2001

    2001

  52. [60]

    Feynman, R.B

    R.P. Feynman, R.B. Leighton, and M. Sands. The Feynman lectures in Physics, Vol. I, II, III . Addison-Wesley, New York, 1963

    1963

  53. [61]

    M. W. Zemansky. Heat and thermodynamics . McGraw-Hill, New-York, 1957

    1957

  54. [62]

    J. Uffink. Boltzmann’s work in statistical physics. In Edw ard N. Zalta, edi- tor, The Stanford Encyclopedia of Philosophy . The Stanford Encyclopedia of Philosophy, winter 2008 edition, 2008

    2008

  55. [63]

    Darrigol

    O. Darrigol. Atoms, Mechanics, and Probability . Oxford University Press, New York, 2018

    2018

  56. [64]

    Gallavotti

    G. Gallavotti. Chaotic hypothesis: Onsager reciprocit y and fluctuation–dissi- pation theorem. Journal of Statistical Physics , 84:899–926, 1996

    1996

  57. [65]

    S. Nos´ e. A unified formulation of the constant temperatu re molecular dynamics methods. Journal of Chemical Physics , 81:511–519, 1984

    1984

  58. [66]

    W. Hoover. Canonical equilibrium phase-space distribu tions. Physical Review A, 31:1695–1697, 1985

    1985

  59. [67]

    D. J. Evans and G. P. Morriss. Statistical Mechanics of Nonequilibrium Fluids . Academic Press, New-York, 1990

    1990

  60. [68]

    Williams, D.J

    S.R. Williams, D.J. Searles, and D.J. Evans. Independen ce of the transient fluctuation theorem to thermostatting details. Physical Review E , 70:066113 (+6), 2004

    2004

  61. [69]

    Choquard

    Ph. Choquard. Lagrangian formulation of nos´ e-hoover a nd of isokinetic dy- namics. www.esi.ac.at, 1996

    1996

  62. [70]

    Gallavotti

    G. Gallavotti. Entropy, Thermostats and Chaotic Hypoth esis. Chaos, 16:043114 (+6), 2006

    2006

  63. [71]

    R. Becker. Electromagnetic fields and interactions . Blaisdell, New-York, 1964

    1964

  64. [72]

    N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai . Steady state elec- tric conductivity in the periodic Lorentz gas. Communications in Mathematical Physics, 154:569–601, 1993

    1993

  65. [73]

    Garrido and G

    P. Garrido and G. Gallavotti. Boundary dissipation in a d riven hard disk system. Journal of Statistical Physics , 126:1201–1207, 2007

    2007

  66. [74]

    J. P. Eckmann and D. Ruelle. Ergodic theory of chaos and st range attractors. Reviews of Modern Physics , 57:617–656, 1985

    1985

  67. [75]

    D. Ruelle. Smooth dynamics and new theoretical ideas in n on-equilibrium statistical mechanics. Journal of Statistical Physics , 95:393–468, 1999

    1999

  68. [76]

    D. Ruelle. What are the measures describing turbulence. Progress in Theoret- ical Physics Supplement , 64:339–345, 1978

    1978

  69. [77]

    D. Ruelle. Measures describing a turbulent flow. Annals of the New York Academy of Sciences , 357:1–9, 1980

    1980

  70. [78]

    Gallavotti and E

    G. Gallavotti and E. Presutti. Thermodynamic limit for i sokinetic thermostats. Journal of Mathematical Physics , 51:0353303 (+9), 2010

    2010

  71. [79]

    D. J. Evans and S. Sarman. Equivalence of thermostatted n onlinear responses. Physical Review E , 48:65–70, 1993

    1993

  72. [80]

    Gallavotti and E

    G. Gallavotti and E. Presutti. Fritionless thermostats and intensive constants of motion. Journal of Statistical Physics , 139:618–629, 2010

    2010

  73. [81]

    Ya. G. Sinai. Markov partitions and C-diffeomorphisms. Functional Analysis and its Applications , 2(1):64–89, 1968

    1968

  74. [82]

    R. Bowen. Markov partitions for axiom A diffeomorphisms. American Journal of Mathematics , 92:725–747, 1970. References 237

    1970

  75. [83]

    Bowen and D

    R. Bowen and D. Ruelle. The ergodic theory of axiom A flows. Inventiones Mathematicae, 29:181–205, 1975

    1975

  76. [84]

    R. Bowen. Markov partitions and minimal sets for axiom A d iffeomorphisms. American Journal of Mathematics , 92:907–918, 1970

    1970

  77. [85]

    D. Ruelle. A measure associated with axiom A attractors. American Journal of Mathematics , 98:619–654, 1976

    1976

  78. [86]

    Gallavotti and D

    G. Gallavotti and D. Cohen. Dynamical ensembles in noneq uilibrium statistical mechanics. Physical Review Letters , 74:2694–2697, 1995

    1995

  79. [87]

    D. Ruelle. Turbulence, strange attractors and chaos . World Scientific, New- York, 1995

    1995

  80. [88]

    D. Ruelle. Positivity of entropy production in nonequil ibrium statistical me- chanics. Journal of Statistical Physics , 85:1–25, 1996

    1996

  81. [89]

    Levesque and N

    D. Levesque and N. Sourlas. Time irreversibility in Stat istical Mechanics. arXiv:2402.12910v3, pages 1–3, 2024

    2024 arXiv

  82. [90]

    Levesque and L

    D. Levesque and L. Verlet. Molecular dynamics and time re versibility. Journal of Statistical Physics , 72:519–537, 1993

    1993

  83. [91]

    J. L. Lebowitz. Boltzmann’s entropy and time’s arrow. Physics Today , September:32–38, 1993

    1993

  84. [92]

    Anosov and Y.G

    D.V. Anosov and Y.G. Sinai. Some smooth ergodic systems. Russian Mathe- matical Surveys , 22:103–167, 1967

    1967

  85. [93]

    Katok and B

    A. Katok and B. Hasselblatt. Introduction to the modern theory of dynami- cal systems , volume 54 of Encyclopedia of Mathematics and its applications . Cambriidge University Press, Cambridge, 1997

    1997

  86. [94]

    D. Ruelle. Elements of differentiable dynamics and bifurcation theory . Aca- demic Press, New-York, 1989

    1989

  87. [95]

    Ya. G. Sinai. Construction of Markov partitions. Functional Analysis and its Applications, 2(3):70–80, 1968

    1968

  88. [96]

    R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeormo r- phisms, volume 470 of Lecture Notes in Mathematics . Springer-Verlag, Berlin- Heidelberg, 1975

    1975

  89. [97]

    Katok and B

    A. Katok and B. Hasselblatt. Introductionto the modern theory of dynami- cal systems , volume 54 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, UK, 1995

    1995

  90. [98]

    Gallavotti

    G. Gallavotti. New methods in nonequilibrium gases and fl uids. Open Systems and Information Dynamics , 6:101–136, 1999

    1999

  91. [99]

    Gallavotti

    G. Gallavotti. Heat and fluctuations from order to chaos. European Physics Journal B, EPJB , 61:1–24, 2008

    2008

  92. [100]

    D. Ruelle. Statistical mechanics of one–dimensional l attice gas. Communica- tions in Mathematical Physics , 9:267–278, 1968

    1968

  93. [101]

    Jiang and Ya

    M. Jiang and Ya. B. Pesin. Equilibrium measures for coup led map lattices: Existence, uniqueness and finite-dimensional approximati ons. Communications in Mathematical Physics , 193:675–711, 1998

    1998

  94. [102]

    Gallavotti

    G. Gallavotti. Entropy production in nonequilibrium s tationary states: a point of view. Chaos, 14:680–690, 2004

    2004

  95. [103]

    Einstein

    E. Einstein. Zur Theorie des Radiometers. Annalen der Physik , 69:241–254, 1922

    1922

  96. [104]

    L. Andrej. The rate of entropy change in non–Hamiltonia n systems. Physics Letters, 111A:45–46, 1982

    1982

  97. [105]

    Kernigham and D.M

    B.W. Kernigham and D.M. Ritchie. The C Programming Language. Prentice Hall Software Series. Prentice Hall, Engelwood Cliffs, N.J. , 1988. References 238

    1988

  98. [106]

    Ornstein

    D. Ornstein. Ergodic Theory, randomness and dynamical systems , volume 5 of Yale Mathematical Monographs. Yale University Press, New Haven, 1974

    1974

  99. [107]

    Gallavotti

    G. Gallavotti. Ising model and Bernoulli shifts. Communications in Mathe- matical Physics , 32:183–190, 1973

    1973

  100. [108]

    Ledrappier

    F. Ledrappier. Mesure d’equilibre sur un reseau. Communications in Mathe- matical Physics , 33:119–128, 1973

    1973

  101. [109]

    Monroy and L

    G. Monroy and L. Russo. A family of codes between some mar kov and bernoulli schemes. Communications in Mathematical Physics , 43:155–159, 1975

    1975

  102. [110]

    Bonetto and G

    F. Bonetto and G. Gallavotti. Reversibility, coarse gr aining and the chaoticity principle. Communications in Mathematical Physics , 189:263–276, 1997

    1997

  103. [111]

    Gallavotti

    G. Gallavotti. Breakdown and regeneration of time reve rsal symmetry in nonequilibrium statistical mechanics. Physica D , 112:250–257, 1998

    1998

  104. [112]

    Dressler

    U. Dressler. Symmetry property of the Lyapunov exponen ts of a class of dissi- pative dynamical systems with viscous damping. Physical Review A , 38:2103– 2109, 1988

    1988

  105. [113]

    Dettman and G

    C. Dettman and G. Morriss. Proof of conjugate pairing fo r an isokinetic ther- mostat. Physical Review E , 53:5545–5549, 1996

    1996

  106. [114]

    Gallavotti

    G. Gallavotti. Topics in chaotic dynamics. Lecture Notes in Physics, ed. Garrido–Marro, 448:271–311, 1995

    1995

  107. [115]

    Constantin and C

    P. Constantin and C. Foias. Global Lyapunov Exponents, Kaplan-Yorke Formulas and the Dimension of the Attractors for 2D Navier-S tokes Equa- tions. Retrieved from the University of Minnesota Digital Conserv ancy, http://hdl.handle.net/11299/4254:1–47, 1983

    1983

  108. [116]

    Bonetto, G

    F. Bonetto, G. Gallavotti, and P. Garrido. Chaotic prin ciple: an experimental test. Physica D , 105:226–252, 1997

    1997

  109. [117]

    Ya. G. Sinai. Gibbs measures in ergodic theory. Russian Mathematical Surveys, 27:21–69, 1972

    1972

  110. [118]

    Ya.G. Sinai. Lectures in ergodic theory. Lecture notes in Mathematics. Prince- ton University Press, Princeton, 1977

    1977

  111. [119]

    Ya.G. Sinai. Topics in ergodic theory , volume 44 of Princeton Mathematical Series. Princeton University Press, 1994

    1994

  112. [120]

    Minlos and J.G

    R.A. Minlos and J.G. Sinai. The phenomenon of phase sepa ration at low temperatures in some lattice models of a gas, i. Math. USSR Sbornik , 2:335– 395, 1967

    1967

  113. [121]

    Gallavotti and D

    G. Gallavotti and D. Cohen. Dynamical ensembles in stat ionary states. Journal of Statistical Physics , 80:931–970, 1995

    1995

  114. [122]

    Gallavotti

    G. Gallavotti. Reversible Anosov diffeomorphisms and l arge deviations. Math- ematical Physics Electronic Journal (MPEJ) , 1:1–12, 1995

    1995

  115. [123]

    Gallavotti and D

    G. Gallavotti and D. Cohen. Note on nonequilibrium stat ionary states and entropy. Physical Review E , 69:035104 (+4), 2004

    2004

  116. [124]

    N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sina i. Derivation of Ohm’s law in a deterministic mechanical model. Physical Review Letters , 70:2209–2212, 1993

    1993

  117. [125]

    Gallavotti

    G. Gallavotti. Fluctuation patterns and conditional r eversibility in nonequi- librium systems. Annales de l’ Institut H. Poincar´ e , 70:429–443, 1999 and chao-dyn/9703007

    1999 arXiv

  118. [126]

    Gallavotti

    G. Gallavotti. Foundations of Fluid Dynamics . (second printing) Springer Verlag, Berlin, 2005

    2005

  119. [127]

    de Groot and P

    S. de Groot and P. Mazur. Non equilibrium thermodynamics . Dover, Mineola, NY, 1984. References 239

    1984

  120. [128]

    Gomez-Marin, J.M.R

    A. Gomez-Marin, J.M.R. Parondo, and C. Van den Broeck. T he footprints of irreversibility. European Physics Letters, 82:5002+4, 2008

    2008

  121. [129]

    Onsager and S

    L. Onsager and S. Machlup. Fluctuations and irreversib le processes. Physical Review, 91:1505–1512, 1953

    1953

  122. [130]

    Machlup and L

    S. Machlup and L. Onsager. Fluctuations and irreversib le processes. ii. Physical Review, 91:1512–1515, 1953

    1953

  123. [131]

    Hurtado, C

    P. Hurtado, C. P´ eres-Espigares, J. Pozo, and P. Garrid o. Symmetries in fluc- tuations far from equilibrium. Proceedings of tha National Academy of Science (PNAS), 108:7704–7709, 2011

    2011

  124. [132]

    Gallavotti

    G. Gallavotti. Extension of Onsager’s reciprocity to l arge fields and the chaotic hypothesis. Physical Review Letters , 77:4334–4337, 1996

    1996

  125. [133]

    Gallavotti and D

    G. Gallavotti and D. Ruelle. SRB states and nonequilibr ium statistical mechan- ics close to equilibrium. Communications in Mathematical Physics , 190:279– 285, 1997

    1997

  126. [134]

    D. Ruelle. Entropy production in nonequilibrium stati stical mechanics. Com- munications in Mathematical Physics , 189:365–371, 1997

    1997

  127. [135]

    Gallavotti

    G. Gallavotti. A local fluctuation theorem. Physica A , 263:39–50, 1999

    1999

  128. [136]

    Pesin and Y.G

    Y.B. Pesin and Y.G. Sinai. Space–time chaos in chains of weakly inteacting hyperbolic mappimgs. Advances in Soviet Mathematics , 3:165–198, 1991

    1991

  129. [137]

    Bricmont and A

    J. Bricmont and A. Kupiainen. High temperature expansi ons and dynamical systems. Communications in Mathematical Physics , 178:703–732, 1996

    1996

  130. [138]

    Bricmont and A

    J. Bricmont and A. Kupiainen. Coupled analytic maps. Nonlinearity, 8:379–396, 1995

    1995

  131. [139]

    Bricmont and A

    J. Bricmont and A. Kupiainen. Infinite dimensional srb m easures. Physica D , 103:18–33, 1997

    1997

  132. [140]

    Gallavotti

    G. Gallavotti. Equivalence of dynamical ensembles and Navier Stokes equa- tions. Physics Letters A , 223:91–95, 1996

    1996

  133. [141]

    Bonetto, G

    F. Bonetto, G. Gallavotti, and G. Gentile. A fluctuation theorem in a random environment. Ergodic Theory and Dynamical Systems , 28:21–47, 2008

    2008

  134. [142]

    S. Olla. Large Deviations for Gibbs Random Fields. Probability Theory and Related Fields, 77:343–357, 1988

    1988

  135. [143]

    Gallavotti, J

    G. Gallavotti, J. L. Lebowitz, and V. Mastropietro. Lar ge deviations in rarefied quantum gases. Journal of Statistical Physics , 108:831–861, 2002

    2002

  136. [144]

    G Gallavotti and F. Perroni. An experimental test of the local fluctuation theorem in chains of weakly interacting anosov s ystems. https://arxiv.org/abs/chao-dyn/9909007, pages 1–15, 1999

    1999 arXiv

  137. [145]

    Gallavotti

    G. Gallavotti. Chaotic hypothesis and universal large deviations properties. Documenta Mathematica, extra volume ICM98, vol. I:205–233, 1998

    1998

  138. [146]

    Mauri, R

    F. Mauri, R. Car, and E. Tosatti. Canonical Statistical Averages of Coupled Quantum-Classical Systems. Europhysics Letters, 24:431–436, 1993

    1993

  139. [147]

    Alonso, A

    J.L. Alonso, A. Castro, J. Clemente-Gallardo, J.C. Cuc hi, P. Echenique, and F. Falceto. Statistics and Nos´ e formalism for Ehrenfest dy namics. Journal of Physics A , 44:395004, 2011

    2011

  140. [148]

    Gallavotti

    G. Gallavotti. The Elements of Mechanics (II edition) . Gallavotti website, Roma, 2008 [I edition was Springer 1984]

    2008

  141. [149]

    Strocchi

    F. Strocchi. Complex Coordinates and Quantum Mechanic s. Reviews of Mod- ern Physics , 38:36–40, 1966

    1966

  142. [150]

    M. Campisi. Quantum Fluctuation Relations for Ensembl es of Wave Functions. New Journal of Physics , 15:1–12, 2013. References 240

    2013

  143. [151]

    F. Seitz. The modern theory of solids . Dover, Mineola, 1987 (reprint)

    1987

  144. [152]

    B. D. Holian, W. G. Hoover, and H. A. W. Posch. Resolution of loschmidt paradox: The origin of irreversible behavior in reversible atomistic dynamics. Physical Review Letters , 59:10–13, 1987

    1987

  145. [153]

    D. Ruelle. A remark on the equivalence of isokinetic and isoenergetic ther- mostats in the thermodynamic limit. Journal of Statistical Physics , 100:757– 763, 2000

    2000

  146. [154]

    Gallavotti

    G. Gallavotti. Nonequilibrium and Fluctuation Relati on. Journal of Statistical Physics, 180:172–226, 2019

    2019

  147. [155]

    Gallavotti

    G. Gallavotti. Reversible viscosity and Navier–Stoke s fluids. Springer Proceed- ings in Mathematics & Statistics , 282:569–580, 2019

    2019

  148. [156]

    Gallavotti

    G. Gallavotti. Viscosity, Reversibility, Chaotic Hyp othesis, Fluctuation Theo- rem and Lyapunov Pairing. Journal of Statistical Physics , 185:21:1–19, 2021

    2021

  149. [157]

    Gallavotti

    G. Gallavotti. Irreversibility time scale. Chaos, 16:023130 (+7), 2006

    2006

  150. [158]

    Gallavotti

    G. Gallavotti. Nonequilibrium statistical mechanics (stationary): Overview. Encyclopedia of Mathematical Physics, ed. J.P. Fran¸ coise , G.L. Naber, Tsou Sheung Tsun , 3:530–539, 2006

    2006

  151. [159]

    D. Ruelle. Differentiation of SRB states. Communications in Mathematical Physics, 187:227–241, 1997

    1997

  152. [160]

    Bonetto, G

    F. Bonetto, G. Gentile, and V. Mastropietro. Electric fi elds on a surface of constant negative curvature. Ergodic Theory and Dynamical Systems , 20:681– 686, 2000

    2000

  153. [161]

    She and E

    Z.S. She and E. Jackson. Constrained Euler system for Na vier-Stokes turbu- lence. Physical Review Letters , 70:1255–1258, 1993

    1993

  154. [162]

    P. Sagaut. Large Eddy Simulation for Incompressible Flows . Scientific compu- tation. Springer, Berlin, 2006

    2006

  155. [163]

    Germano, U

    M. Germano, U. Piomell, P. Moin, and W.H. Cabot. A dynami c subgridscale eddy viscosity model. Physics of Fluids A , 3:1760–1766, 1991

    1991

  156. [164]

    Fefferman

    C. Fefferman. Existence & smoothness of the Navier–Stokes equation . in J.Carlson,A.Jaffe,A.Wiles: The millennium prize problems . American Math- ematical Society, Providence, RI, 2006

    2006

  157. [165]

    Gallavotti

    G. Gallavotti. Dynamical ensembles equivalence in flui d mechanics. Physica D, 105:163–184, 1997

    1997

  158. [166]

    Constantin

    P. Constantin. On the euler equations of incompressibl e fluids. Bulletin of the American Mathematical Society , 44:603–621, 207

  159. [167]

    Margazoglu, L

    G. Margazoglu, L. Biferale, M. Cencini, G. Gallavotti, and V. Lucarini. Non- equilibrium Ensembles for the 3D Navier-Stokes Equations. Physical Review E, 105:065110, 2022

    2022

  160. [168]

    Shukla, B

    V. Shukla, B. Dubrulle, S. Nazarenko, G. Krstulovic, an d S. Thalabard. Phase transition in time-reversible Navier-Stokes equations. Physical Review E , 100:043104, 2019

    2019

  161. [169]

    Seshasayanan, S

    K. Seshasayanan, S. Eswaran, M. Maji, S. Ghosh, and V. Sh ukla. Equivalence of nonequilibrium ensembles:turbulence with a dual cascad e. Physical Review E, 108:015102, 2021

    2021

  162. [170]

    S. Kuksin. The k41 theory and turbulence in 1d burgers eq uation. Chaos, 34:1–29, 2024

    2024

  163. [171]

    Biferale, M

    L. Biferale, M. Cencini, M. DePietro, G. Gallavotti, an d V. Lucarini. Equiva- lence of non-equilibrium ensembles in turbulence models. Physical Review E , 98:012201, 2018. References 241

    2018

  164. [172]

    Pikeroen, A

    Q. Pikeroen, A. Barral, G. Costa, C. Campolina, A. Maily baev, and B. Dubrulle. Tracking complex singularities of fluids on log -lattices. arXiv.2312.01702, 2023

    2023 arXiv

  165. [173]

    Gallavotti and V

    G. Gallavotti and V. Lucarini. Equivalence of Non-Equi librium Ensembles and Representation of Friction in Turbulent Flows: The Lorenz 9 6 Model. Journal of Statistical Physics , 156:1027–1053, 2014

    2014

  166. [174]

    Gallavotti

    G. Gallavotti. Intermittency and time arrow in statist ical mechanics and tur- bulence. Fields Institute Communications , 31:165–172, 2002

    2002

  167. [175]

    Gallavotti

    G. Gallavotti. Microscopic chaos and macroscopic entr opy in fluids. Journal of Statistical Mechanics (JSTAT) , 2006:P10011 (+9), 2006

    2006

  168. [176]

    Pomeau and P

    Y. Pomeau and P. Manneville. Different ways to turbulenc e in dissipative dynamical systems. Physica D , 1:219–226, 1980

    1980

  169. [177]

    Gallavotti

    G. Gallavotti. Ensembles, Turbulence and Fluctuation Theorem. European Physics Journal E , 43:37, 2020

    2020

  170. [178]

    J. Kurchan. Fluctuation theorem for stochastic dynami cs. Journal of Physics A, 31:3719–3729, 1998

    1998

  171. [179]

    Lebowitz and H

    J. Lebowitz and H. Spohn. A Gallavotti–Cohen type symme try in large devia- tion functional for stochastic dynamics. Journal of Statistical Physics , 95:333– 365, 1999

    1999

  172. [180]

    C. Maes. The fluctuation theorem as a Gibbs property. Journal of Statistical Physics, 95:367–392, 1999

    1999

  173. [181]

    Collet, H

    P. Collet, H. Epstein, and G. Gallavotti. Perturbation s of geodesic flows on surfaces of constant negative curvature and their mixing pr operties. Commu- nications in Mathematical Physics , 95:61–112, 1984

    1984

  174. [182]

    Iacobucci

    A. Iacobucci. Nonequilibrium stationary states of rotor and oscillator c hains. https://theses.hal.science/tel-01792503, Paris, 2018

    2018

  175. [183]

    Iacobucci, S

    A. Iacobucci, S. Olla, and G. Stoltz. Convergence rates for nonequilibrium langevin dynamics. Annales Math´ ematiques du Qu´ ebec, 43:73–98, 2018

    2018

  176. [184]

    Van Zon and E

    R. Van Zon and E. G. D. Cohen. Extension of the fluctuation theorem. Physical Review Letters, 91:110601 (+4), 2003

    2003

  177. [185]

    Van Zon and E

    R. Van Zon and E. G. D. Cohen. Extended heat-fluctuation t heorems for a system with deterministic and stochastic forces. Physical Review E , 69:056121 (+14), 2004

    2004

  178. [186]

    Joubaud, N.B

    S. Joubaud, N.B. Garnier, and S. Ciliberto. Fluctuatio n theorems for harmonic oscillators. Journal of Statistical Mechanics , 9:P09018, 2007

    2007

  179. [187]

    D. J. Evans, D. J. Searles, and L. Rondoni. Application o f the Gallavotti–Cohen fluctuation relation to thermostated steady states near equ ilibrium. Physical Review E , 71:056120 (+12), 2005

    2005

  180. [188]

    F. Zamponi. Is it possible to experimentally verify the fluctuation relation? a review of theoretical motivations and numerical evidence. Journal of Statistical Mechanics, page P02008, 2007

    2007

  181. [189]

    Cugliandolo, J

    L.F. Cugliandolo, J. Kurchan, and L. Peliti. Energy flow , partial equilibration, and effective temperatures in systems with slow dynamics. Physical Review E , pages 2898–3914, 1997

    1997

  182. [190]

    Crisanti and F

    A. Crisanti and F. Ritort. Violation of the fluctuation- dissipation theorem in glassy systems: basic notions and the numerical evidence. Journal of Physics A, pages R181–R290, 2003

    2003

  183. [191]

    Nakabeppu and T

    O. Nakabeppu and T. Suzuki. Microscale temperature mea surement by scan- ning thermal microscopy. Journal of Thermal Analysis and Calorimetry , 69:727–737, 2002. References 242

    2002

  184. [192]

    D.T. Smith. A square root circuit to linearize feedback in temperature con- trollers. Journal of Physics E: Scientific Instruments , 5:528, 1972

    1972

  185. [193]

    Feitosa and N

    K. Feitosa and N. Menon. A fluidized granular medium as an instance of the fluctuation theorem. Physical Review Letters , 92:164301+4, 2004

    2004

  186. [194]

    Bonetto, G

    F. Bonetto, G. Gallavotti, A. Giuliani, and F. Zamponi. Fluctuations relation and external thermostats: an application to granular mater ials. Journal of Statistical Mechanics, 2006(05):P05009, 2006

    2006

  187. [195]

    Puglisi, P

    A. Puglisi, P. Visco, A. Barrat, E. Trizac, and F. van Wij land. Fluctuations of internal energy flow in a vibrated granular gas. Physical Review Letters (cond-mat/0509105), 95:110202 (+4), 2005

    2005 arXiv

  188. [196]

    E. L. Grossman, E.L., Tong Zhou, and E. Ben-Naim. Toward s granular hydro- dynamics in two-dimensions. cond-mat/9607165, 1996

    1996 arXiv

  189. [197]

    J. Brey, M. J. Ruiz-Montero, and F. Moreno. Boundary con ditions and normal state for a vibrated granular fluid. Physical Review E , 62:5339–5346, 2000

    2000

  190. [198]

    Bonetto, N

    F. Bonetto, N. Chernov, and J. L. Lebowitz. (global and l ocal) fluctuations in phase-space contraction in deteministic stationary non equilibrium. Chaos, 8:823–833, 1998

    1998

  191. [199]

    Bonetto and J

    F. Bonetto and J. L. Lebowitz. Thermodynamic entropy pr oduction fluctuation in a two-dimensional shear flow model. Physical Review E , 64:056129, 2001

    2001

  192. [200]

    Zamponi, G

    F. Zamponi, G. Ruocco, and L. Angelani. Fluctuations of entropy production in the isokinetic ensemble. Journal of Statistical Physics , 115:1655–1668, 2004

    2004

  193. [201]

    Giuliani, F

    A. Giuliani, F. Zamponi, and G. Gallavotti. Fluctuatio n relation beyond linear response theory. Journal of Statistical Physics , 119:909–944, 2005

    2005

  194. [202]

    Visco, A

    P. Visco, A. Puglisi, A. Barrat, E. Trizac, and F. van Wij land. Fluctuations of power injection in randomly driven granular gases. Journal of Statistical Physics, 125:529–564, 2005

    2005

  195. [203]

    Jepps, D

    O. Jepps, D. Evans, and D. Searles. The fluctuation theor em and Lyapunov weights. Physica D , 187:326–337, 2004

    2004

  196. [204]

    G. Zeuner. Grundgz¨ uge der Mechanischen W¨ armetheorie. Buchandlung J.G. Engelhardt, Freiberg, 1860

  197. [205]

    M.A. Masson. Sur la corr´ elation des propri´ et´ es physiques des corps. Annales de Chimie , 53:257–293, 1858

  198. [206]

    Gallavotti

    G. Gallavotti. Ergodicity: a historical perspective. equilibrium and nonequi- librium. European Physics Journal H , 41,:181–259, 2016

    2016

  199. [207]

    Gallavotti and I

    G. Gallavotti and I. Jauslin. A Theorem on Ellipses, an I ntegrable System and a Theorem of Boltzmann. arXiv, 2008.01955:1–9, 2020

    2008 arXiv

  200. [208]

    G. Felder. Poncelet Property and Quasi-periodicity of the Integrable Boltz- mann System. Letters in Mathematical Physics , 111:1–19, 2021

    2021

  201. [209]

    Landau and E.M

    L.D. Landau and E.M. Lifschitz. M´ ecanique des fluides. MIR, Moscow, 1971

    1971

  202. [210]

    Clausius

    R. Clausius. Ueber eine ver¨ anderte form des zweiten ha uptsatzes der mecha- nischen w¨ armetheorie.Annalen der Physik und Chemie , 93:481–506, 1854

  203. [211]

    Clausius

    R. Clausius. On the application of the theorem of the equ ivalence of transfor- mations to interior work. Philosophical Magazine , 4-XXIV:81–201, 1862

  204. [212]

    Love Letters

    J. Renn. Einstein’s controversy with Drude and the orig in of statistical me- chanics: a new glimpse from the “Love Letters”. Archive for the history of exact sciences, 51:315–354, 1997

    1997

  205. [213]

    J.C. Maxwell. On the dynamical theory of gases. Philosophical Magazine , XXXV:129–145, 185–217, 1868. References 243

  206. [214]

    G. E. Uhlenbeck. An outline of Statistical Mechanics, in Fundamental proble ms in Statistical Mechanics, II . ed. E. G. D. Cohen, North Holland, Amsterdam, 1968

    1968

  207. [215]

    Helmholtz

    H. Helmholtz. Prinzipien der Statik monocyklischer Systeme , volume III of Wissenschaftliche Abhandlungen . Barth, Leipzig, 1895

  208. [216]

    Helmholtz

    H. Helmholtz. Studien zur Statik monocyklischer Systeme , volume III of Wis- senschaftliche Abhandlungen . Barth, Leipzig, 1895

  209. [217]

    Marchioro and E

    C. Marchioro and E. Presutti. Thermodynamics of partic le systems in pres- ence of external macroscopic fields. Communications in Mathematical Physics , 27:146–154, 1972

    1972

  210. [218]

    Whittaker

    E.T. Whittaker. A treatise on the analytic dynamics of particles & rigid bodi es. Cambridge University Press, Cambridge, 1917 (reprinted 19 89)

    1917

  211. [219]

    Arnold and A

    V.I. Arnold and A. Avez. Ergodic probems of Classical Mechanics . Mathemat- ical Physics Monographs. Benjamin, 1968

    1968

  212. [220]

    S. Smale. Differentiable dynamical systems. Bullettin of the American Math- ematical Society, 73:747–818, 1967

    1967

  213. [221]

    Hofer and E.Zehnder

    H. Hofer and E.Zehnder. Symplectic invariants and Hamiltonian dynamics . Birkhauser Advanced Texts. Birkhauser Verlag, Basel, 1994

    1994

  214. [222]

    Wojtkowski and C

    M.P. Wojtkowski and C. Liverani. Conformally symplect ic dynamics and sym- metry of the lyapunov spectrum. Communications in Mathematical Physics , 194:47–60, 1998

    1998

  215. [223]

    Evans, EGD Cohen, and G

    D. Evans, EGD Cohen, and G. Morriss. Viscosity of a simpl e fluid from its maximal lyapunov exponents. Physical Review A , 42:5990–5997, 1990

    1990

  216. [224]

    Benettin, L

    G. Benettin, L. Galgani, A. Giorgilli, and J. Strelcyn. Lyapunov charateris- tic exponents for smooth dynamical systems and for Hamilton ian systems; A method for computing all of them. Part 2, Numerical applicat ion. Meccanica, 15:21–30, 1980

    1980

  217. [225]

    Gallavotti

    G. Gallavotti. Navier-stokes equation: irreversibil ity turbulence and ensembles equivalence. arXiv:1902.09610, page 09160, 2019

    1902 arXiv

  218. [226]

    D. Ruelle. Large volume limit of the distribution of cha racteristic exponents in turbulence. Communications in Mathematical Physics , 87:287–302, 1982

    1982

  219. [227]

    E. Lieb. On characteristic exponents in turbulence. Communications in Math- ematical Physics , 92:473–480, 1984

    1984

  220. [228]

    Gallavotti

    G. Gallavotti. Nonequilibrium and irreversibility . Theoretical and Mathemat- ical Physics. Springer-Verlag, 2014

    2014

  221. [229]

    Gallavotti, L

    G. Gallavotti, L. Rondoni, and E. Segre. Lyapunov spect ra and nonequilibrium ensembles equivalence in 2d fluid. Physica D , 187:358–369, 2004

    2004

  222. [230]

    Minlos and J.G

    R.A. Minlos and J.G. Sinai. The phenomenon of phase sepa ration at low temperatures in some lattice models of a gas, ii. Transactions of the Moscow Mathematical Society, 19:121–196, 1968

    1968

  223. [231]

    Gallavotti and S

    G. Gallavotti and S. Miracle-Sol´ e. Absence of phase tr ansitions in hard-core one dimensional systems with long range interactions. Journal of Mathematical Physics, 11:147–154, 1970

    1970

  224. [232]

    Cassandro and E

    M. Cassandro and E. Olivieri. Renormalization group an d analyticity in one dimension: A proof of dobrushin’s theorem. Communications in Mathematical Physics, 80:255–269, 1981

    1981

  225. [233]

    Gallavotti

    G. Gallavotti. Chaotic dynamics, fluctuations, non-eq uilibrium ensembles. Chaos, 8:384–392, 1998

    1998

  226. [234]

    Jarzynski

    C. Jarzynski. Nonequilibrium equality for free energy difference. Physical Review Letters, 78:2690–2693, 1997. Subject index 244

    1997

  227. [235]

    Jarzynski

    C. Jarzynski. Hamiltonian derivation of a detailed fluc tuation theorem. Journal of Statistical Physics , 98:77–102, 1999

    1999

  228. [236]

    Evans and D

    D.J. Evans and D. Searles. Equilibrium microstates whi ch generate second law violating steady state. Physical Review E , 50:1645–1648, 1994

    1994

  229. [237]

    G. N. Bochkov and Yu. E. Kuzovlev. Nonlinear fluctuation -dissipation rela- tions and stochastic models in nonequilibrium thermodynam ics: I. generalized fluctuation-dissipation theorem. Physica A , 106:443–479, 1981

    1981

  230. [238]

    G. N. Bochkov and Yu. E. Kuzovlev. Nonlinear fluctuation -dissipation re- lations and stochastic models in nonequilibrium thermodyn amics: II. Kinetic potential and variational principles for nonlinear irreve rsible processes. Physica A, 106:480–520, 1981. List of frequent abbr...

    1981

  231. [239]

    102 126 128 130 15 155 16 18 187 192 204 205 206 207 208 22 228 42 5 66 7 73 86 95 97 IX

  232. [240]

    138 14 197 199 202 5 6

  233. [241]

    11 12 137 161 204 205 251 6 7 8

  234. [242]

    149 15 16 171 176 184 50 7 9

  235. [243]

    149 16 168 196 197 199 200 251 7

  236. [244]

    10 13 138 204 251 8 9

  237. [245]

    13 14 15 154 16 167 171 175 176 251 8

  238. [246]

    15 175 181 192 194 207 251 8

  239. [247]

    13 154 155 172 179 183 205 251 46 47 68 8 9

  240. [248]

    11 13 176 183 187 251 8

  241. [249]

    12 146 154 187 251 46 9

  242. [250]

    11 13 16 176 18 183 185 19 20 203 204 205 251 48 50 68

  243. [251]

    15 210 51 63 64 65 66 70 76 92 93

  244. [252]

    26 31 33 Citations index 246

  245. [253]

    214 216 29 36 37 38 41 42 65 71 83 84 87 88

  246. [254]

    210 34 36 50 52 55 59 82

  247. [255]

    103 Citations index 247

  248. [256]

    solution of a mechanical problem

    231 Chapters abstracts 248 R Chapters abstracts Chapter 1 A brief survey of the development of Equilibrium theory, from Carno t’s theorem to the least action principle and the Ergodic Hypothesis (EH ), as a background for the study of stationary Non Equilibrium. Particular att...

    1971

  249. [257]

    Heat theorem

    Heat theorem, action principle, EH . The first Boltzmann’s work on the second principle mechanical basis (the “Heat theorem”), [29]

  250. [258]

    ensemble

    Collision analysis and equipartition . By a combinatorial argument, in pres- ence of hard core (or instantaneous) interactions, Maxwell’s distr ibution is derived and the microcanonical “ensemble” is introduced, [43]

  251. [259]

    An attempt to give an example of an er- godic system

    Solution of a mechanical problem . An attempt to give an example of an er- godic system. Comments on the relation, and conflict, with KAM theo ry, [47] . 4,5) Recurrence and periodicity: Clausius version of the heat theorem, [34] . 6,7) Priority discussions Boltzmann-Clausius o...

  252. [260]

    Collisions between molecules imply that the molecule atoms visit all points of the single molecule phase space; EH is n ot used

    On the ergodic hypothesis . Collisions between molecules imply that the molecule atoms visit all points of the single molecule phase space; EH is n ot used. And microcanonical and canonical ensembles are introduced : for the first time the the microcanonical distribution unique...

  253. [261]

    u nder an hypothesis

    Canonical ensemble and ergodic hypothesis . Solution for the thermal equi- librium for the molecules of a gas with a finite number of point masses “u nder an hypothesis”. The hypothesis concerns the uniqueness of the s tationary dis- tribution, raised already in 8). And ergodic...

  254. [262]

    Heat theorem without dynamics . It is shown how the hypothesis that, assuming that the equilibrium distribution of the entire system is the m icro- canonical one, then defining the heat dQ received by the body as the variation of the total average energy dE plus the work dW don...

  255. [263]

    Boltzmann’s sea

    Irreversibility: Loschmidt and “Boltzmann’s sea” . “ We want to prove the second theorem in mechanical terms, founding it on the nature of t he inter- action laws and without imposing any restriction on the initial condition s, knowledge of which is not supposed ”, [44]

  256. [264]

    ratios of the number of different states of which we want to compute the probabilities

    Discrete phase space, count of its points and entropy , The relation between the theorems on the mechanical theory of heat and on the probab ility theory of equipartition and problem of determining the “ ratios of the number of different states of which we want to compute the p...

  257. [265]

    Monocyclic and orthodic systems. Ensembles : “ The most complete proof of the second main theorem is manifestly based on the remark that, for each given mechanical system, equations that are analogous to equatio ns of the theory of heat hold ”, [41]

  258. [266]

    Indeed the properties of a body supposed to be a uniform plenum may be affirmed dogmatically, but cannot be explained mathema ti- cally

    Maxwell 1866: “ Indeed the properties of a body supposed to be a uniform plenum may be affirmed dogmatically, but cannot be explained mathema ti- cally”, [33] . Chapters abstracts 254 Chapter 7 (Appendices) A) A simple example of the mechanical interpretation of heat theore m B)...

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