The fluctuation relation (FR) asserts that in a nonequilibrium steady state the probability of entropy-producing fluctuations exceeds that of entropy-consuming ones by an exponential factor fixed by the entropy production itself, giving the quantitative microscopic form of the second law. For deterministically thermostatted systems the statement is a theorem only for idealized uniformly hyperbolic (Anosov) dynamics and only for the phase-space contraction rate; its transfer to the physically measurable entropy production (heat delivered to the reservoirs divided by their temperatures) rests on two unproven pillars: Gallavotti's Chaotic Hypothesis, and the boundedness of the total-derivative correction that relates phase-space contraction to entropy production. The source paper (Gallavotti, Chaos 16, 043114 (2006)) additionally conjectured that finite thermostat containers may be replaced by infinite Hamiltonian reservoirs without changing the results, and acknowledged these gaps. Later work proved the finite-time, trajectory-level part of the thermostat equivalence for restricted potentials and dimensions, exhibited genuine failures of the naive heat fluctuation theorem in stochastic models with unbounded correction terms, and tested the relation numerically and experimentally; but the stationary-state equivalence, a rigorous FR for the entropy production, and the asymptotics of the associated Bonetto integral identity remain open. The problem is to settle this cluster rigorously.

Background

A nonequilibrium steady state of a molecular system - a state with stationary statistics but nonzero entropy production, maintained by external driving - is commonly modeled in molecular dynamics by deterministic "thermostatted" equations of motion: forces perform work on a test system, and the unmodeled environment is replaced by thermostat regions whose energy is controlled by friction-like feedback forces. In the isokinetic thermostat, obtained from Gauss' principle of least constraint, the friction force is chosen at each instant to keep the kinetic energy of each thermostat region constant. The dynamics so defined is reversible (it commutes with velocity inversion) but non-Hamiltonian, and its phase-space volume contracts: the phase-space contraction rate is σ(x)=−div fE(x)\sigma(x)=-\mathrm{div}\,f_E(x), minus the divergence of the equations of motion fEf_E, the subscript marking the applied forcing.

Gallavotti's Chaotic Hypothesis (CH) proposes that the motions on the attracting set of a chaotic system can be regarded, for the purpose of computing averages, as those of a transitive Anosov system - the mathematical model of fully developed, uniformly hyperbolic chaos. Anosov systems possess a natural invariant probability distribution, the SRB (Sinai-Ruelle-Bowen) distribution, which describes the statistics of the trajectories of almost every initial condition and is smooth along the unstable (expanding) directions. For reversible Anosov systems the Gallavotti-Cohen fluctuation theorem (FT) is a theorem: writing pp for the time average of σ\sigma over an observation time τ\tau, normalized by its SRB average σ+\sigma_+, the large-deviation rate function ζ(p)\zeta(p) - governing the exponential decay, as τ→∞\tau\to\infty, of the probability of observing the value pp - satisfies the symmetry ζ(−p)=ζ(p)−pσ+\zeta(-p)=\zeta(p)-p\sigma_+ on the interval where both signs of pp occur.

The experimentally accessible quantity, however, is not σ\sigma but the entropy production rate ε(x)=∑iQi/(kBTi)\varepsilon(x)=\sum_i Q_i/(k_B T_i), where QiQ_i is the heat per unit time extracted by thermostat ii at temperature TiT_i. In the standard geometry of a test system between finite thermostat containers, σ\sigma and ε\varepsilon differ by a total time derivative,

σ(x)=ε(x)+R˙(x),\sigma(x)=\varepsilon(x)+\dot R(x),

where RR is a phase function built from the thermostat kinetic energies and the interaction energies across the thermostat boundaries. If RR is bounded, the time averages of σ\sigma and ε\varepsilon coincide as τ→∞\tau\to\infty, and the FT transfers to ε\varepsilon; for physical (non-Anosov) systems this transfer is called the fluctuation relation (FR), the word "relation" marking that it is a consequence of the unproven CH rather than a theorem. The transfer is what makes the FR testable: ε\varepsilon depends only on quantities local to the system-thermostat boundaries, not on the bulk of the thermostat model. An integral consequence is Bonetto's formula, ⟨e−∫0τε(Stx) dt⟩SRB=eO(1)\langle e^{-\int_0^\tau \varepsilon(S_t x)\,dt}\rangle_{\mathrm{SRB}}=e^{O(1)}, where StS_t is the time-tt map and the angle brackets denote the SRB average; the mnemonic limit 11 as τ→∞\tau\to\infty would be exact only if the FT held at finite τ\tau.

In the paper "Entropy, thermostats, and chaotic hypothesis" (Chaos 16, 043114 (2006)) - the source paper - Gallavotti consolidated this framework and explicitly flagged its open foundations: the boundedness of RR is "a strong assumption", tied to the unanalyzed "efficiency" of thermostats, whose violation "may lead to interesting consequences"; the conjecture that the finite thermostat containers can be replaced by infinite containers initially in equilibrium Gibbs states - the probability distributions describing thermal equilibrium at temperature TiT_i - is one "whose proof seems quite far"; the FR for ε\varepsilon rests on the CH, itself a hypothesis rather than a theorem; and the FR is only conjectured for non-reversible (for instance viscous, −νiX˙i-\nu_i\dot X_i) thermostat forces. Within the same conditional framework, the corrections to, and rate of approach in, Bonetto's formula remain undetermined. The problem is the rigorous settlement of this cluster. Comprehensive reviews of the deterministic-thermostat program and its open issues are given by Jepps and Rondoni (J. Phys. A 43, 133001 (2010)) and in Gallavotti's later accounts cited below.

Problem Statement

For reversible deterministically thermostatted particle systems of the source's class - a test system driven by external forces, coupled to finite thermostat containers regulated by Gaussian isokinetic or isoenergetic (constant-energy) friction - settle the rigorous status of the fluctuation relation for the physical entropy production, together with the thermostat-equivalence statements it presupposes. Concretely:

(1a) Fluctuation relation for ε\varepsilon: prove, or disprove by an explicit counterexample within the class, that the large-deviation rate function of the dimensionless entropy production p=(1/τ)∫0τε(Stx)/ε+ dtp=(1/\tau)\int_0^\tau \varepsilon(S_t x)/\varepsilon_+\,dt (with ε+\varepsilon_+ its SRB average) satisfies ζ(−p)=ζ(p)−pε+\zeta(-p)=\zeta(p)-p\varepsilon_+ on the fluctuation interval, under precisely identified hypotheses.

(1b) Unbounded correction term: determine whether and how failure of boundedness of the correction term RR (unbounded interaction potentials, thermostat boundary kinetic-energy terms) modifies the symmetry of (1a) and its finite-time (finite-τ\tau) corrections.

(2) Thermostat equivalence at stationarity: prove, or disprove, that the stationary statistics of test-system observables - including the fluctuation statistics of ε\varepsilon - coincide when heat is extracted by finite isokinetic or isoenergetic thermostats and when the thermostat containers are replaced by infinite reservoirs initially in Gibbs states at temperatures TiT_i; that is, extend the established finite-time (trajectory-level) equivalence to the t→∞t\to\infty stationary limit, including the treatment of unbounded (Lennard-Jones-type) and hard-core potentials.

(3) Bonetto's formula: determine the asymptotic behavior of ⟨e−∫0τε(Stx) dt⟩SRB\langle e^{-\int_0^\tau \varepsilon(S_t x)\,dt}\rangle_{\mathrm{SRB}} as τ→∞\tau\to\infty - whether it converges to 1, at what rate, and with what corrections - or prove that it remains bounded without converging.

These three components jointly support one objective: the rigorous foundation of the claim that the measurable entropy production of a thermostatted nonequilibrium system obeys the fluctuation relation. Component (2) establishes that the thermostatted ε\varepsilon is the entropy production of the corresponding Hamiltonian-reservoir system; component (1) establishes the fluctuation symmetry itself and the effect of an unbounded correction term; component (3) establishes its integral form and finite-time corrections. Resolution standard: all three components are required, and resolving any one of them does not by itself resolve the task - in particular, a single explicit counterexample within the class settles sub-deliverable (1a) only, so (1b), (2) and (3) remain to be settled. Within (1a) the hypotheses are to be identified and stated by the solver, and a result proved only under hypotheses that exclude part of the source's class counts as partial, with the excluded part remaining open.

References

  1. G. Gallavotti, "Entropy, thermostats, and chaotic hypothesis", Chaos 16, 043114 (2006). DOI: 10.1063/1.2372713; arXiv:cond-mat/0606690. https://doi.org/10.1063/1.2372713
  2. D. J. Evans, E. G. D. Cohen, G. P. Morriss, "Probability of second law violations in shearing steady states", Phys. Rev. Lett. 71, 2401 (1993). DOI: 10.1103/PhysRevLett.71.2401. https://doi.org/10.1103/PhysRevLett.71.2401
  3. G. Gallavotti, E. G. D. Cohen, "Dynamical Ensembles in Nonequilibrium Statistical Mechanics", Phys. Rev. Lett. 74, 2694 (1995). DOI: 10.1103/PhysRevLett.74.2694. https://doi.org/10.1103/PhysRevLett.74.2694
  4. R. van Zon, E. G. D. Cohen, "Extended heat-fluctuation theorems for a system with deterministic and stochastic forces", Phys. Rev. E 69, 056121 (2004). DOI: 10.1103/PhysRevE.69.056121; arXiv:cond-mat/0311357. https://doi.org/10.1103/PhysRevE.69.056121
  5. T. Taniguchi, E. G. D. Cohen, "Nonequilibrium Steady State Thermodynamics and Fluctuations for Stochastic Systems", J. Stat. Phys. 130, 633 (2007). DOI: 10.1007/s10955-007-9471-1; arXiv:0708.2940. https://doi.org/10.1007/s10955-007-9471-1
  6. F. Bonetto, G. Gallavotti, A. Giuliani, F. Zamponi, "Chaotic Hypothesis, Fluctuation Theorem and singularities", J. Stat. Phys. 123, 39 (2006). DOI: 10.1007/s10955-006-9047-5; arXiv:cond-mat/0507672. https://doi.org/10.1007/s10955-006-9047-5
  7. C. Mejía-Monasterio, L. Rondoni, "On the Fluctuation Relation for Nosé-Hoover Boundary Thermostated Systems", J. Stat. Phys. 133, 617 (2008). DOI: 10.1007/s10955-008-9618-8; arXiv:0710.3673. https://doi.org/10.1007/s10955-008-9618-8
  8. O. G. Jepps, L. Rondoni, "Deterministic thermostats, theories of nonequilibrium systems and parallels with the ergodic condition", J. Phys. A: Math. Theor. 43, 133001 (2010). DOI: 10.1088/1751-8113/43/13/133001. https://doi.org/10.1088/1751-8113/43/13/133001
  9. G. Gallavotti, "On Thermostats: Isokinetic or Hamiltonian? Finite or Infinite?", Chaos 19, 013101 (2009). DOI: 10.1063/1.3054710; arXiv:0810.1510. https://doi.org/10.1063/1.3054710
  10. G. Gallavotti, E. Presutti, "Nonequilibrium, thermostats, and thermodynamic limit", J. Math. Phys. 51, 015202 (2010). DOI: 10.1063/1.3257618; arXiv:0905.3150. https://doi.org/10.1063/1.3257618
  11. G. Gallavotti, E. Presutti, "Thermodynamic limit for isokinetic thermostats", J. Math. Phys. 51, 053303 (2010). DOI: 10.1063/1.3376659; arXiv:0908.3060. https://doi.org/10.1063/1.3376659
  12. G. Gallavotti, E. Presutti, "Frictionless thermostats and intensive constants of motion", J. Stat. Phys. 139, 618 (2010). DOI: 10.1007/s10955-010-9949-0; arXiv:0907.1188. https://doi.org/10.1007/s10955-010-9949-0
  13. K. K. Lin, L.-S. Young, "Nonequilibrium Steady States for Certain Hamiltonian Models", J. Stat. Phys. 139, 630 (2010). DOI: 10.1007/s10955-010-9958-z; arXiv:1004.0412. https://doi.org/10.1007/s10955-010-9958-z
  14. G. Gallavotti, "Nonequilibrium and Irreversibility", Springer, Theoretical and Mathematical Physics (2014). ISBN: 978-3-319-06757-5; DOI: 10.1007/978-3-319-06758-2; arXiv:1311.6448. https://doi.org/10.1007/978-3-319-06758-2
  15. G. Carinci, C. Giardinà, E. Presutti, "Stationary States in Infinite Volume with Non-zero Current", J. Stat. Phys. 180, 366 (2019). DOI: 10.1007/s10955-019-02427-9; arXiv:1906.06120. https://doi.org/10.1007/s10955-019-02427-9
  16. G. Gallavotti, "Nonequilibrium and Fluctuation Relation", J. Stat. Phys. 180, 172 (2019). DOI: 10.1007/s10955-019-02376-3; arXiv:1906.10069. https://doi.org/10.1007/s10955-019-02376-3
  17. F. Bonetto, N. Chernov, A. Korepanov, J. L. Lebowitz, "Autonomous evolution of electron speeds in a thermostatted system: exact results", Nonlinearity 32, 2055 (2019). DOI: 10.1088/1361-6544/ab005d; arXiv:1809.00016. https://doi.org/10.1088/1361-6544/ab005d
  18. G. Gallavotti, "Finite thermostats in classical and quantum nonequilibrium", Eur. Phys. J. Spec. Top. 227, 217 (2018). DOI: 10.1140/epjst/e2018-700096-x; arXiv:2111.04078. https://doi.org/10.1140/epjst/e2018-700096-x
  19. G. Gallavotti, "Viscosity, Reversibillity [sic; the journal record misspells 'reversibility'], Chaotic Hypothesis, Fluctuation Theorem and Lyapunov Pairing", J. Stat. Phys. 185, 21 (2021). DOI: 10.1007/s10955-021-02830-1; arXiv:2102.10125. https://doi.org/10.1007/s10955-021-02830-1
  20. G. Margazoglou, L. Biferale, M. Cencini, G. Gallavotti, V. Lucarini, "Nonequilibrium ensembles for the three-dimensional Navier-Stokes equations", Phys. Rev. E 105, 065110 (2022). DOI: 10.1103/PhysRevE.105.065110; arXiv:2201.00530. https://doi.org/10.1103/PhysRevE.105.065110
  21. M. M. Bandi, J. R. Cressman, W. I. Goldburg, "Test of the Fluctuation Relation in Lagrangian Turbulence on a Free Surface", J. Stat. Phys. 130, 27 (2007). DOI: 10.1007/s10955-007-9355-4. https://doi.org/10.1007/s10955-007-9355-4
  22. F. Zonta, S. Chibbaro, "Entropy production and Fluctuation Relation in turbulent thermal convection", EPL 114, 50011 (2016). DOI: 10.1209/0295-5075/114/50011. https://doi.org/10.1209/0295-5075/114/50011