Pith. sign in

REVIEW 4 minor 77 references

Fluctuation theorems hold for energy exchange with a finite-inertia work source once its back-action entropy is included.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-10 16:53 UTC pith:CMD62DK6

load-bearing objection Clean autonomous versions of the standard classical FTs, with explicit back-action and a controlled M o∞ recovery of the agent-driven results.

arxiv 2607.07843 v1 pith:CMD62DK6 submitted 2026-07-08 cond-mat.stat-mech

Fluctuation theorems for autonomous work

Christopher Jarzynski , Sebastian Deffner , Saar Rahav This is my paper
classification cond-mat.stat-mech
keywords fluctuation theoremsautonomous workstochastic thermodynamicsback-actionJarzynski equalityCrooks relationentropy productionnonequilibrium processes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Standard fluctuation theorems for work assume an external agent drives a system by changing a parameter on a fixed schedule. This paper derives the corresponding equalities when the system instead swaps energy with a physical work source that itself moves under the joint dynamics. The work source feels a fluctuating back-action, so its own entropy change must be tracked; once that term is kept, the familiar integral and detailed theorems reappear. In the limit of infinite inertia the back-action vanishes and the ordinary non-autonomous statements are recovered. The results therefore show that the second-law equalities of stochastic thermodynamics apply to fully autonomous energy exchange, not only to externally scripted protocols.

Core claim

For a system S and reversible work source R that evolve together under Hamiltonian or Markovian stochastic dynamics from an initial state in which S is conditionally equilibrated given R’s coordinate, the integral fluctuation theorem ⟨exp(−β(W−ΔF)−Δφ)⟩=1 holds, where W is the energy lost by R, ΔF is the free-energy change of S evaluated at the initial and final coordinates of R, and Δφ is the stochastic entropy change of R. Parallel autonomous theorems are obtained for exclusive work, total entropy production, and Crooks’s detailed relation; each reduces to its non-autonomous counterpart when R’s mass tends to infinity.

What carries the argument

The autonomous integral fluctuation theorem ⟨e^{−β(W−ΔF)−Δφ}⟩=1, obtained by averaging over trajectories of the joint phase-space density that begins as ρ0(X,V)π(z|X) and evolves under Liouville or the corresponding Markov generator. The extra factor e^{−Δφ} accounts for the back-action entropy of the work source and is what allows the equality to close.

Load-bearing premise

The system of interest must start in the conditional equilibrium distribution fixed by the work source’s initial coordinate; if that preparation fails, the equalities as written no longer hold.

What would settle it

Simulate or measure the joint trajectories of a finite-mass work source coupled to a small system prepared out of conditional equilibrium; if the average of exp(−β(W−ΔF)−Δφ) systematically deviates from 1 while the dynamics remain Hamiltonian or Markovian, the central claim is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any experimental platform that can prepare a work source with tunable inertia can continuously interpolate between autonomous and non-autonomous fluctuation theorems.
  • The second-law inequality for mean work acquires an additive term equal to the Shannon-entropy change of the work source, which is negligible only for macroscopic sources.
  • Exclusive and inclusive definitions of autonomous work each support their own fluctuation theorem, exactly as they do in the non-autonomous setting.
  • Crooks’s detailed relation survives for autonomous processes once forward and reverse ensembles are defined by suitably related initial distributions of the work source.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bookkeeping of back-action entropy should convert existing quantum fluctuation theorems that treat work reservoirs into fully autonomous statements.
  • Feedback-control equalities may be re-derived by identifying the controller with the finite-inertia work source, thereby unifying measurement-and-feedback thermodynamics with ordinary energy exchange.
  • Exact solvable models with one degree of freedom for R and few for S would allow direct numerical tests of the finite-M corrections without approximation.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The manuscript derives classical fluctuation theorems for autonomous work, in which a system of interest S exchanges energy with a reversible work source R under joint Hamiltonian or Markovian stochastic dynamics, without external control of a work parameter. The central integral result is ⟨e^{-β(W-ΔF)-Δφ}⟩=1 (Eqs. 3, 15, 30), where W is the energy lost by R, ΔF=F(X_τ)-F(X_0) is a stochastic free-energy change of S, and Δφ is the stochastic entropy change of R arising from back-action. Parallel autonomous theorems are obtained for exclusive work (Eq. 44), total entropy production (Eq. 50), and a Crooks-type detailed relation (Eq. 65). Each reduces to its standard non-autonomous counterpart in the infinite-inertia limit M o∞ with delta-distributed initial conditions for R. Derivations are given for isolated RS, for RS coupled to a heat bath (implicitly via stochastic dynamics and explicitly in the SI), and under both inclusive and exclusive energy partitions.

Significance. If correct, the results place the principal classical work and entropy-production fluctuation theorems on a fully autonomous footing that accounts for back-action on the work source. This closes a conceptual gap between textbook thermodynamics (energy exchange between physical bodies) and the agent-driven protocols of stochastic thermodynamics, and supplies a controlled limit (M o∞) in which the two pictures coincide. The derivations are short, rest only on Liouville/propagator conservation of the equilibrium measure and the stated initial conditions, and are corroborated by three independent routes (Hamiltonian, stochastic, explicit-bath SI). The framework also opens natural connections to feedback control and bipartite systems. These strengths make the paper a clear contribution to the foundations of stochastic thermodynamics.

minor comments (4)
  1. The preparation assumption that S begins in conditional equilibrium π(z|X) (Eqs. 10, 42, 58) is load-bearing for the integral work theorems and is stated by analogy with the non-autonomous Jarzynski setup. A short explicit remark that this is the natural autonomous counterpart of the usual initial-equilibrium assumption, and that the total-entropy-production theorem (Eq. 50) already covers arbitrary initial S states, would help readers unfamiliar with the literature.
  2. In the Crooks section the ordering of limits (first M o∞, then ε o0 for the family of initial distributions) is essential to avoid divergences in the logarithmic terms of Eqs. 59. A single clarifying sentence would make this technical point more transparent.
  3. The Discussion sketches experimental routes (macroscopic Hamiltonian systems; optical-trap feedback that emulates Newton’s law for X). Expanding one of these sketches by a few sentences would strengthen the paper’s connection to possible tests without altering the theoretical claims.
  4. Notation for the exclusive free energy F_0 (Eq. 43) and the inclusive free energy F(X) (Eq. 9) is clear once introduced, but a brief reminder when both appear in the same paragraph would reduce the chance of momentary confusion.

Circularity Check

0 steps flagged

No significant circularity: autonomous FTs are derived from Liouville/propagator conservation plus explicit initial-condition and work definitions, not forced by construction or self-citation.

full rationale

The central equalities (Eqs. 15, 30, 44, 50, 65) follow by direct substitution of the definitions of W (energy lost by R), ΔF = F(X au)−F(X0), and Δφ (Shannon entropy change of R) into the ensemble average, followed by the equilibrium-preserving property of the dynamics (Eqs. 20, 23, 27) and Liouville/propagator change of variables. The initial conditional equilibrium of S (Eqs. 10, 42) is an explicit modeling assumption, stated by analogy with the non-autonomous Jarzynski setup and not derived from the target equality; when it is relaxed the paper supplies the natural generalization (total-entropy-production FT, Eq. 50). The M o∞ reduction to the non-autonomous theorems is a controlled limit argument, not a definitional identity. Self-citations (e.g., Deffner–Jarzynski 2013) supply only background inequalities and do not underwrite the new equalities. No parameters are fitted, no uniqueness theorem is imported, and no known empirical pattern is merely renamed. The derivation chain is therefore self-contained against its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The paper is a first-principles derivation within classical stochastic thermodynamics. It imports standard dynamical assumptions (Hamiltonian flow, Markovian generators that preserve conditional equilibrium, detailed balance, time-reversal invariance of HS) and the usual initial-equilibrium preparation of S. No free parameters are fitted. No new particles or forces are postulated; the work source R is an ordinary classical degree of freedom. The only modeling choices that are paper-specific are the inclusive/exclusive energy split and the definition of autonomous work as energy lost by R.

axioms (6)
  • domain assumption RS (or RST) evolves under Hamiltonian dynamics generated by HRS (or HRST), or under Markovian stochastic dynamics whose generator preserves the conditional equilibrium of S (Eqs. 7, 20–23).
    Standard classical mechanics / stochastic thermodynamics; load-bearing for Liouville and propagator identities used in every derivation.
  • domain assumption Initial state of S is the conditional equilibrium π(z|X) (inclusive) or π0(z) (exclusive); R’s initial distribution ρ0 is arbitrary (Eqs. 10, 42).
    Direct analogue of the non-autonomous Jarzynski initial-equilibrium assumption; required for the cancellation that yields ⟨e^{−β(W−ΔF)−Δφ}⟩=1.
  • domain assumption For the Crooks theorem: HS is time-reversal invariant and the propagator satisfies detailed balance (Eqs. 56–57).
    Standard microscopic reversibility assumptions used in classical detailed fluctuation theorems.
  • domain assumption Work source R does not couple directly to the heat bath T; only S does (SI Hamiltonian HRST).
    Standard idealization of a reversible work source; stated explicitly in the Supporting Information.
  • domain assumption Inclusive vs exclusive partitioning of interaction energy Hint into S or R is a valid thermodynamic convention (Eqs. 5, 38–39).
    Follows Jarzynski’s earlier inclusive/exclusive framework for non-autonomous work; both conventions yield consistent but different FTs.
  • standard math In the infinite-inertia limit, Xt = X0 + V0 t and Vt = V0 with no back-action (Eq. 31), and delta-distributed initial conditions for R eliminate fluctuations in ΔF.
    Direct consequence of Newton’s law as M→∞; used to recover non-autonomous theorems.

pith-pipeline@v1.1.0-grok45 · 26648 in / 3286 out tokens · 134473 ms · 2026-07-10T16:53:42.210160+00:00 · methodology

0 comments
read the original abstract

Classical fluctuation theorems for work have been obtained theoretically, and verified experimentally, within a non-autonomous framework in which work is performed on a system of interest, ${\cal S}$, by the external manipulation of a work parameter, such as a piston's position. Here we obtain fluctuation theorems within an autonomous framework in which ${\cal S}$ exchanges energy with a reversible work source, ${\cal R}$. The two subsystems, ${\cal R}$ and ${\cal S}$, interact with one another as they evolve under Hamiltonian or stochastic dynamics, without external intervention. In this setting, we must account for the back-action of ${\cal S}$ on ${\cal R}$, which is absent in the non-autonomous setting. We obtain autonomous versions of standard fluctuation theorems for work and entropy production. In each case, we argue, the autonomous fluctuation theorem reduces to its non-autonomous counterpart when ${\cal R}$'s inertia becomes infinitely large.

Figures

Figures reproduced from arXiv: 2607.07843 by Christopher Jarzynski, Saar Rahav, Sebastian Deffner.

Figure 1
Figure 1. Figure 1: (b). In this notion of work, which we call non￾autonomous, there is no explicit reference to the physical body R. In its place, it is convenient to imagine an external agent who controls how the work parameter, λ, varies with time. For macroscopic systems the laws of thermodynamics apply to both autonomous and non-autonomous work, and there is little need to distinguish between the two notions. (a) (b) [P… view at source ↗
Figure 2
Figure 2. Figure 2: Toy model illustrating our general setup. S consists of three particles of mass m, and R is a particle of mass M. The wall at the left is fixed, and springs denote harmonic couplings. ∗ Following the literature, we distinguish between integral fluctuation theorems, expressed as averaged exponentials (Eq. 1), and detailed fluctuation theorems, expressed as ratios of distributions (Eq. 52). The system and wo… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages · 5 internal anchors

  1. [2]

    (Scribner’s, New Y ork), (1902)

    JW Gibbs, Elementary Principles in Statistical Mechanics. (Scribner’s, New Y ork), (1902)

    work page 1902

  2. [3]

    (Wiley, New Y ork, USA), (1985)

    H Callen, Thermodynamics and an Introduction to Thermostastistics. (Wiley, New Y ork, USA), (1985)

    work page 1985

  3. [4]

    J Anacleto, Work reservoirs in thermodynamics. Eur. J. Phys.31, 617–624 (2010)

    work page 2010

  4. [5]

    (Elsevier, Amsterdam) Vol

    L Landau, E Lifshitz, A Course of Theoretical Physics. (Elsevier, Amsterdam) Vol. 3, (1980)

    work page 1980

  5. [6]

    (Chapman and Hall), 2nd edition, (1993)

    CBP Finn, Thermal Physics. (Chapman and Hall), 2nd edition, (1993)

    work page 1993

  6. [7]

    U Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys.75, 126001 (2012)

    work page 2012

  7. [8]

    (Princeton University Press, Princeton), (2021)

    L Peliti, S Pigolotti, Stochastic Thermodynamics: An Introduction. (Princeton University Press, Princeton), (2021)

    work page 2021

  8. [9]

    (Oxford University Press, Oxford), (2024)

    DT Limmer, Statistical Mechanics and Stochastic Thermodynamics. (Oxford University Press, Oxford), (2024)

    work page 2024

  9. [10]

    (Cambridge University Press, Cambridge), (2025)

    U Seifert, Stochastic Thermodynamics. (Cambridge University Press, Cambridge), (2025)

    work page 2025

  10. [11]

    C Jarzynski, Nonequilibrium equality for free energy differences. Phys. Rev. Lett.78, 2690 – 2693 (1997)

    work page 1997

  11. [12]

    C Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach. Phys. Rev. E56, 5018 – 5035 (1997)

    work page 1997

  12. [13]

    (Oxford University Press, New Y ork), (1987) Sec

    D Chandler, Introduction to Modern Statistical Mechanics. (Oxford University Press, New Y ork), (1987) Sec. 5.5

    work page 1987

  13. [14]

    GN Bochkov, YE Kuzovlev, General theory of thermal fluctuations in nonlinear systems. Sov. Phys. JETP45, 125–130 (1977)

    work page 1977

  14. [15]

    generalized fluctuation-dissipation theorem

    GN Bochkov, YE Kuzovlev, Nonlinear fluctuation–dissipation relations and stochastic models in nonequilibrium thermodynamics I. generalized fluctuation-dissipation theorem. Phys. A 106, 443–479 (1981)

    work page 1981

  15. [16]

    GE Crooks, Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys.90, 1481–1487 (1998)

    work page 1998

  16. [17]

    GE Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E60, 2721–2726 (1999)

    work page 1999

  17. [18]

    G Hummer, A Szabo, Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proc. Natl. Acad. Sci.98, 3658 (2001)

    work page 2001

  18. [19]

    U Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett.95, 040602 (2005)

    work page 2005

  19. [20]

    S Deffner, C Jarzynski, Information processing and the second law of thermodynamics: An inclusive, Hamiltonian approach. Phys. Rev. X3, 041003 (2013)

    work page 2013

  20. [21]

    C Jarzynski, Microscopic analysis of Clausius-Duhem processes. J. Stat. Phys.96, 415 – 427 (1999)

    work page 1999

  21. [22]

    Comptes Rendus Physique 8, 495–506 (2007)

    C Jarzynski, Comparison of far-from-equilibrium work relations. Comptes Rendus Physique 8, 495–506 (2007)

    work page 2007

  22. [23]

    A Sone, S Deffner, Jarzynski equality for conditional stochastic work. J. Stat. Phys.183, 11 (2021)

    work page 2021

  23. [24]

    U Seifert, First and second law of thermodynamics at strong coupling. Phys. Rev. Lett.116, 020601 (2016)

    work page 2016

  24. [25]

    P Talkner, P H ¨anggi, Open system trajectories specify fluctuating work but not heat. Phys. Rev. E94, 022143 (2016)

    work page 2016

  25. [26]

    H Miller, J Anders, Entropy production and time asymmetry in the presence of strong interactions. Phys. Rev. E95(2017)

    work page 2017

  26. [27]

    (Addison-Wesley, Reading, Massachusetts), 2nd edition, (1980)

    H Goldstein, Classical Mechanics. (Addison-Wesley, Reading, Massachusetts), 2nd edition, (1980)

    work page 1980

  27. [28]

    Acta Phys

    C Jarzynski, Equilibrium free energies from nonequilibrium processes. Acta Phys. Pol.29, 1609–1622 (1998)

    work page 1998

  28. [29]

    DJ Evans, EGD Cohen, GP Morriss, Probability of second law violations in shearing steady states. Phys. Rev. Lett.71, 2401 (1993)

    work page 1993

  29. [30]

    DJ Evans, DJ Searles, Equilibrium microstates which generate second law violating steady states. Phys. Rev. E50, 1645 (1994)

    work page 1994

  30. [31]

    G Gallavotti, EGD Cohen, Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett.74, 2694 (1995)

    work page 1995

  31. [32]

    G Gallavotti, EGD Cohen, Dynamical ensembles in stationary states. J. Stat. Phys.80, 931 (1995)

    work page 1995

  32. [33]

    J Kurchan, Fluctuation theorem for stochastic dynamics. J. Phys. A: Math. Gen.31, 3719–3729 (1998)

    work page 1998

  33. [34]

    JL Lebowitz, H Spohn, A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys.95, 333–365 (1999)

    work page 1999

  34. [35]

    C Maes, The fluctuation theorem as a Gibbs property. J. Stat. Phys.95, 367–392 (1999)

    work page 1999

  35. [36]

    Jarzynski Relations for Quantum Systems and Some Applications

    H Tasaki, Jarzynski relations for quantum systems and some applications. arXiv:cond-mat/0009244v2 (2000)

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  36. [37]

    A Quantum Fluctuation Theorem

    J Kurchan, A quantum fluctuation theorem. arXiv:cond-mat/0007360v2 (2000)

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  37. [38]

    S Mukamel, Quantum extension of the Jarzynski relation: Analogy with stochastic dephasing. Phys. Rev. Lett.90, 170604 (2003)

    work page 2003

  38. [39]

    M Esposito, U Harbola, S Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys.81, 1665–1702 (2009)

    work page 2009

  39. [40]

    M Campisi, P H¨anggi, P Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys.83, 771–791 (2011)

    work page 2011

  40. [41]

    S Deffner, JP Paz, WH Zurek, Quantum work and the thermodynamic cost of quantum measurements. Phys. Rev. E94, 010103 (2016)

    work page 2016

  41. [42]

    K Micadei, GT Landi, E Lutz, Quantum fluctuation theorems beyond two-point measurements. Phys. Rev. Lett.124, 090602 (2020)

    work page 2020

  42. [43]

    K Maeda, T Holdsworth, S Deffner, A Sone, Detailed fluctuation theorem from the one-time measurement scheme. Phys. Rev. A108, L050203 (2023)

    work page 2023

  43. [44]

    M Campisi, P Talkner, P H¨anggi, Fluctuation Theorem for Arbitrary Open Quantum Systems. Phys. Rev. Lett.102, 210401 (2009)

    work page 2009

  44. [45]

    EPL (Europhys

    S Deffner, M Brunner, E Lutz, Quantum fluctuation theorems in the strong damping limit. EPL (Europhys. Lett.)94, 30001 (2011)

    work page 2011

  45. [46]

    S Deffner, E Lutz, Nonequilibrium entropy production for open quantum systems. Phys. Rev. Lett.107, 140404 (2011)

    work page 2011

  46. [47]

    D Kafri, S Deffner, Holevo’s bound from a general quantum fluctuation theorem.Phys. Rev. A 86, 044302 (2012)

    work page 2012

  47. [48]

    AE Rastegin, Non-equilibrium equalities with unital quantum channels. J. Stat. Mech.: Theor. Exp. p. P06016 (2013)

    work page 2013

  48. [49]

    EPL (Europhys

    S Deffner, Quantum entropy production in phase space. EPL (Europhys. Lett.)103, 30001 (2013)

    work page 2013

  49. [50]

    AE Rastegin, K ˙Zyczkowski, Jarzynski equality for quantum stochastic maps. Phys. Rev. E 89, 012127 (2014)

    work page 2014

  50. [51]

    JP Santos, GT Landi, M Paternostro, Wigner entropy production rate. Phys. Rev. Lett.118, 220601 (2017)

    work page 2017

  51. [52]

    F Binder, LA Correa, C Gogolin, J Anders, G Adesso

    HJD Miller, Hamiltonian of Mean Force for Strongly-Coupled Systems in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, Fundamental Theories of Physics, eds. F Binder, LA Correa, C Gogolin, J Anders, G Adesso. (Springer International Publishing, Cham), pp. 531–549 (2018)

    work page 2018

  52. [53]

    P Strasberg, M Esposito, Non-markovianity and negative entropy production rates. Phys. Rev. E99, 012120 (2019)

    work page 2019

  53. [54]

    ´A Rivas, Strong coupling thermodynamics of open quantum systems. Phys. Rev. Lett.124, 160601 (2020)

    work page 2020

  54. [55]

    AVS Quantum Sci.5, 032001 (2023)

    A Sone, DO Soares-Pinto, S Deffner, Exchange fluctuation theorems for strongly interacting quantum pumps. AVS Quantum Sci.5, 032001 (2023)

    work page 2023

  55. [56]

    Universal energy fluctuations in inelastic scattering processes

    SL Jacob, J Goold, GT Landi, F Barra, Universal energy fluctuations in inelastic scattering processes. arXiv:2404.04923v2 [quant-ph] (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  56. [57]

    Work and heat exchanged during sudden quenches of strongly coupled quantum systems

    Z Davoudi, et al., Work and heat exchanged during sudden quenches of strongly coupled quantum systems. arXiv:2502.19418v1 [quant-ph] (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  57. [58]

    D Cohen, Y Imry, Straightforward quantum-mechanical derivation of the Crooks fluctuation theorem and the Jarzynski equality. Phys. Rev. E86, 011111 (2012)

    work page 2012

  58. [59]

    ´AM Alhambra, L Masanes, J Oppenheim, C Perry, Fluctuating work: From quantum thermodynamical identities to a second law equality. Phys. Rev. X6, 041017 (2016)

    work page 2016

  59. [60]

    njp Quantum Inf.4, 59 (2018)

    J Monsel, C Elouard, A Auff `eves, An autonomous quantum machine to learn the thermodynamic arrow of time. njp Quantum Inf.4, 59 (2018)

    work page 2018

  60. [61]

    Quantum5, 408 (2021)

    P Lipka-Bartosik, P Mazurek, M Horodecki, Second law of thermodynamics for batteries with vacuum state. Quantum5, 408 (2021)

    work page 2021

  61. [62]

    J Lyu, AB Boyd, JP Crutchfield, Efficient quantum work reservoir at the nanoscale. Phys. Rev. A110, 012225 (2024)

    work page 2024

  62. [63]

    C Han, D Cohen, E Sela, Quantum limitation on experimental testing of nonequilibrium fluctuation theorems. Phys. Rev. B110, 115153 (2024)

    work page 2024

  63. [64]

    C Han, N Katz, E Sela, Measuring work in quantum many-body systems using a dynamical work agent. Phys. Rev. B112, 035614 (2025)

    work page 2025

  64. [65]

    Quantum measurement of work in mesoscopic systems

    AV Varma, D Cohen, Quantum measurement of work in mesoscopic systems. arXiv:2507.09977v1 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  65. [66]

    T Sagawa, M Ueda, Generalized Jarzynski equality under nonequilibrium feedback control. Phys. Rev. Lett.104, 090602 (2010)

    work page 2010

  66. [67]

    D Hartich, AC Barato, U Seifert, Stochastic thermodynamics of bipartite systems: transfer entropy inequalities and a Maxwell’s demon interpretation.J. Stat. Mech.: Theor. Exp. p. P02016 (2014)

    work page 2014

  67. [68]

    JM Horowitz, M Esposito, Thermodynamics with continuous information flow. Phys. Rev. X4, 031015 (2014)

    work page 2014

  68. [69]

    J Ehrich, DA Sivak, Energy and information flows in autonomous systems. Front. Phys.11, 1108357 (2023)

    work page 2023

  69. [70]

    JG Kirkwood, Statistical mechanics of fluid mixtures. J. Chem. Phys.3, 300–313 (1935)

    work page 1935

  70. [71]

    B Roux, T Simonson, Implicit solvent models. Biophys. Chem.78, 1–20 (1999)

    work page 1999

  71. [72]

    C Jarzynski, Nonequilibrium work theorem for a system strongly coupled to a thermal environment. J. Stat. Mech.: Theor. Exp. p. P09005 (2004)

    work page 2004

  72. [73]

    MF Gelin, M Thoss, Thermodynamics of a subensemble of a canonical ensemble. Phys. Rev. E79, 051121 (2009)

    work page 2009

  73. [74]

    Fluctuation theorems for autonomous work

    C Jarzynski, Stochastic and macroscopic thermodynamics of strongly coupled systems. Phys. Rev. X7, 011008 (2017). Jarzynskiet al. PNAS —Dec. 12, 2025— vol. 122 — no. 51 —9 1 WORK AND HEAT Supporting Information for “Fluctuation theorems for autonomous work” Christopher Jarzynski, Sebastian Deffner and Saar Rahav In the main text, when deriving autonomous ...

    work page 2017

  74. [76]

    Fluctuation theorem for inclusive work We first suppose that initial conditions forRSTare sampled from f0(Γ) =ρ0(X,V) Π(y,z|X),[S15] where ρ0(X,V ) is an arbitrary distribution on R’s phase space, andST is in a state of conditional equilibrium (Eq. S5). Eq. S15 is analogous to Eq. 10 of the main text. From these initial conditions, SRT evolves along a Ham...

    work page

  75. [77]

    Fluctuation theorem for exclusive work Following Eq. 38 of the main text, we partition HS(z;X) into a term that depends only on the microstate of S, and anR-S interaction term: HS(z;X) =H 0 S(z) +H int(z,X).[S19] We then write HRST (Γ) = [M 2 V 2 +H int(z,X) ] + [ H0 S(z) +HT (y) +h int(y,z) ] =HR(X,V;z) +H 0 ST (y,z), [S20] ¶Eq. S14 is equivalent to Eq. ...

    work page 2025

  76. [78]

    S8, let us rewrite Eq

    Fluctuation theorem for total entropy production Using Eq. S8, let us rewrite Eq. S7 as follows: H+ S (z;X) =H S(z;X)−β−1lnZ+ T (z) Z0 T ,[S25] with Z+ T (z) = ∫ dye−β[HT (y)+hint(y,z)].[S26] We further introduce πT (y|z) =1 Z+ T (z) e−β[HT (y)+hint(y,z)],[S27] which is the equilibrium state ofT, conditioned on the microstate ofS. In Sections 2 and 3,R’s ...

    work page doi:10.1073/pnas.2524775122

  77. [79]

    Crooks’s fluctuation theorem for autonomous work As in the main text, we consider two autonomous processes, F and R, corresponding to two choices for initial conditions, fF 0 (Γ) and fR 0 (Γ); see Eq. 58. The microstate Γ now includes the coordinates and momenta of the heat bath, T . We assume that ST begins in a conditional equilibrium state: fF 0 (Γ) =ρ...

    work page 2025