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arxiv: 2606.07263 · v2 · pith:LJSK7DYWnew · submitted 2026-06-05 · ❄️ cond-mat.stat-mech

Oscillatory-nonnormal decomposition of dissipation in Ornstein-Uhlenbeck processes

Ryuna Nagayama , Artemy Kolchinsky , Sosuke Ito This is my paper

Pith reviewed 2026-07-02 22:52 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Ornstein-Uhlenbeck processentropy production rateoscillatory behaviornonnormalitydissipation-coherence trade-offrelaxation trade-offnoise-induced oscillations
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The pith

Entropy production in Ornstein-Uhlenbeck processes decomposes into oscillatory and nonnormal contributions, each linked to its own trade-off.

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

The paper decomposes the steady-state entropy production rate in an Ornstein-Uhlenbeck process into two parts: one from oscillatory behavior and one from nonnormality. The oscillatory part yields a dissipation-coherence trade-off that bounds entropy production per oscillatory period by the number of oscillations in one correlation time, and this bound is twice as strict as prior results. The nonnormal part yields a trade-off between entropy production and the speed of relaxation. This decomposition is shown in a bead-spring model. A sympathetic reader would care because it separates distinct physical sources of dissipation in linear stochastic dynamics.

Core claim

We provide a decomposition of the steady-state entropy production rate associated with an Ornstein-Uhlenbeck process into two contributions: one associated with oscillatory behavior and one associated with nonnormality. We also show that each contribution is associated with a different fundamental trade-off. The oscillatory contribution leads to the dissipation-coherence trade-off for noise-induced oscillations, which bounds the entropy production per oscillatory period by the number of oscillations within one correlation time. Notably, the tradeoff is twice as strict as those conjectured or derived for other systems. The nonnormal contribution leads to a trade-off between entropy production

What carries the argument

The oscillatory-nonnormal decomposition of the steady-state entropy production rate, which separates contributions from oscillatory dynamics and nonnormal matrix properties.

If this is right

  • The oscillatory contribution implies that entropy production per period cannot exceed the number of oscillations possible within the system's correlation time.
  • This oscillatory bound is twice as strict as similar bounds derived for other systems.
  • The nonnormal contribution implies that higher entropy production enables faster relaxation to steady state.
  • The decomposition holds for the simple bead-spring model used as demonstration.

Where Pith is reading between the lines

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

  • This separation could guide the design of stochastic systems to minimize unwanted dissipation while preserving desired oscillations.
  • The approach might extend to other linear stochastic processes where nonnormality and oscillations coexist.
  • Experimental tests in physical systems like colloidal beads could verify the stricter bound on dissipation-coherence.

Load-bearing premise

The steady-state entropy production rate of an Ornstein-Uhlenbeck process can be cleanly split additively into oscillatory and nonnormal contributions that each match independent physical trade-offs.

What would settle it

A numerical simulation of an Ornstein-Uhlenbeck process where the sum of the proposed oscillatory and nonnormal entropy production contributions fails to equal the total computed entropy production rate.

Figures

Figures reproduced from arXiv: 2606.07263 by Artemy Kolchinsky, Ryuna Nagayama, Sosuke Ito.

Figure 1
Figure 1. Figure 1: FIG. 1. Numerical demonstration of the oscillatory-nonnormal de [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. (a) The autocorrelation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical demonstration of the oscillatory-nonnormal de [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

We provide a decomposition of the steady-state entropy production rate associated with an Ornstein-Uhlenbeck process into two contributions: one associated with oscillatory behavior and one associated with nonnormality. We also show that each contribution is associated with a different fundamental trade-off. The oscillatory contribution leads to the dissipation-coherence trade-off for noise-induced oscillations, which bounds the entropy production per oscillatory period by the number of oscillations within one correlation time. Notably, the tradeoff is twice as strict as those conjectured or derived for other systems. The nonnormal contribution leads to a trade-off between entropy production and acceleration of relaxation. We also demonstrate the decomposition using a simple bead-spring model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to decompose the steady-state entropy production rate of an Ornstein-Uhlenbeck process into an oscillatory contribution (from imaginary parts of drift matrix eigenvalues) and a nonnormal contribution (from non-normality of the matrix). Each part is associated with a fundamental trade-off: the oscillatory part yields a dissipation-coherence trade-off bounding entropy production per oscillatory period by the number of oscillations in one correlation time, claimed to be twice as strict as prior results; the nonnormal part yields a trade-off between entropy production and acceleration of relaxation. The decomposition is demonstrated on a bead-spring model.

Significance. If valid, this provides a useful algebraic decomposition of dissipation in linear diffusions based on the Lyapunov equation and standard entropy production formula, allowing separate bounding of contributions from oscillations and non-normality. The stricter trade-off for oscillatory dissipation and the relaxation trade-off could be relevant for systems with noise-induced oscillations or transient non-normal dynamics. The numerical confirmation in the bead-spring example supports the split.

minor comments (1)
  1. [Abstract] Abstract: The statement that the oscillatory tradeoff is 'twice as strict as those conjectured or derived for other systems' would benefit from explicit citation of the compared works and a brief indication of how the factor of two arises in the derivation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the recognition of its potential utility, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; algebraic split from standard OU equations

full rationale

The central decomposition of steady-state entropy production follows directly from the Lyapunov equation for the covariance matrix and the standard expression for entropy production rate in linear diffusions. The oscillatory/nonnormal split is obtained by algebraic rearrangement of these quantities (imaginary parts of eigenvalues vs. non-normality measure of the drift matrix). No fitted parameters are renamed as predictions, no self-citations are load-bearing for the additivity or trade-off bounds, and the bead-spring example serves only as numerical confirmation. The derivation is self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, invented entities, or non-standard axioms. The work rests on the standard definition of Ornstein-Uhlenbeck processes and the existence of a steady-state entropy production rate.

axioms (1)
  • domain assumption Ornstein-Uhlenbeck processes possess a well-defined steady-state entropy production rate that can be decomposed additively.
    Invoked by the opening sentence of the abstract as the basis for the claimed split.

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Works this paper leans on

123 extracted references · 8 canonical work pages · 4 internal anchors

  1. [1]

    The four eigenvalues are expressed with(s1, s2)∈ {+1,−1} 2 as λs1,s2 =k 0 +k int +s 1 r k2 int − (a−b) 2 4 + is2 a+b 2

    This is verified by computing the eigenvalues ofK. The four eigenvalues are expressed with(s1, s2)∈ {+1,−1} 2 as λs1,s2 =k 0 +k int +s 1 r k2 int − (a−b) 2 4 + is2 a+b 2 . (17) Whenb=−a,λ s1,s2 is complex-valued only ifk2 int <(a− b)2/4, so that the square root becomes imaginary. Thus, if the coupling is sufficiently strong (|kint| ≥ |a−b|/2) and b=−a, al...

    2020

  2. [2]

    G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev.36, 823 (1930)

    1930

  3. [3]

    M. C. Wang and G. E. Uhlenbeck, On the theory of the Brow- nian motion II, Rev. Mod. Phys.17, 323 (1945)

    1945

  4. [4]

    Risken,The Fokker-Planck Equation: Methods of Solution and Applications(Springer Berlin Heidelberg, 1996)

    H. Risken,The Fokker-Planck Equation: Methods of Solution and Applications(Springer Berlin Heidelberg, 1996)

    1996

  5. [5]

    Gardiner,Stochastic Methods: A Handbook for the Natural and Social Sciences(Springer Berlin, Heidelberg, 2009)

    C. Gardiner,Stochastic Methods: A Handbook for the Natural and Social Sciences(Springer Berlin, Heidelberg, 2009)

    2009

  6. [6]

    Lax, Fluctuations from the nonequilibrium steady state, Rev

    M. Lax, Fluctuations from the nonequilibrium steady state, Rev. Mod. Phys.32, 25 (1960)

    1960

  7. [7]

    Schmiedl and U

    T. Schmiedl and U. Seifert, Efficiency at maximum power: An analytically solvable model for stochastic heat engines, Europhys. Lett.81, 20003 (2008)

    2008

  8. [8]

    Phys.8, 143 (2012)

    V.BlickleandC.Bechinger,Realizationofamicrometre-sized stochastic heat engine, Nat. Phys.8, 143 (2012)

    2012

  9. [9]

    R.VanZon,S.Ciliberto,andE.Cohen,Powerandheatfluctu- ationtheoremsforelectriccircuits,Phys.Rev.Lett.92,130601 (2004)

    2004

  10. [10]

    Ciliberto, A

    S. Ciliberto, A. Imparato, A. Naert, and M. Tanase, Heat flux andentropyproducedbythermalfluctuations,Phys.Rev.Lett. 110, 180601 (2013)

    2013

  11. [11]

    Chiang, C.-L

    K.-H. Chiang, C.-L. Lee, P.-Y. Lai, and Y.-F. Chen, Electri- cal autonomous Brownian gyrator, Phys. Rev. E96, 032123 (2017)

    2017

  12. [12]

    Melanson, M

    D. Melanson, M. Abu Khater, M. Aifer, K. Donatella, M. Hunter Gordon, T. Ahle, G. Crooks, A. J. Martinez, F. Sbahi, and P. J. Coles, Thermodynamic computing system for AI applications, Nat. Commun.16, 3757 (2025)

    2025

  13. [13]

    F. Mura, G. Gradziuk, and C. P. Broedersz, Nonequilibrium scaling behavior in driven soft biological assemblies, Phys. Rev. Lett.121, 038002 (2018)

    2018

  14. [14]

    Commun.10, 1666 (2019)

    J.Li,J.M.Horowitz,T.R.Gingrich,andN.Fakhri,Quantify- ing dissipation using fluctuating currents, Nat. Commun.10, 1666 (2019)

    2019

  15. [15]

    Van Vu, V

    T. Van Vu, V. T. Vo, and Y. Hasegawa, Entropy production estimation with optimal current, Phys. Rev. E101, 042138 (2020)

    2020

  16. [16]

    1 (Elsevier, 1992)

    N.G.VanKampen,Stochasticprocessesinphysicsandchem- istry, Vol. 1 (Elsevier, 1992). 6

    1992

  17. [17]

    Kubo, The fluctuation-dissipation theorem, Reports on progress in physics29, 255 (1966)

    R. Kubo, The fluctuation-dissipation theorem, Reports on progress in physics29, 255 (1966)

    1966

  18. [18]

    C. Kwon, P. Ao, and D. J. Thouless, Structure of stochastic dynamics near fixed points, Proc. Natl. Acad. Sci. USA102, 13029 (2005)

    2005

  19. [19]

    K.Hasselmann,StochasticclimatemodelspartI.Theory,Tel- lus28, 473 (1976)

    1976

  20. [20]

    Aslyamov, K

    T. Aslyamov, K. Ptaszyński, and M. Esposito, Macroscopic fluctuation-responsetheoryanditsuseforgeneregulatorynet- works, Phys. Rev. Lett.136, 067102 (2026)

    2026

  21. [21]

    K.TomitaandH.Tomita,Irreversiblecirculationoffluctuation, Prog. Theor. Phys.51, 1731 (1974)

    1974

  22. [22]

    P. J. Thomas and B. Lindner, Phase descriptions of a mul- tidimensional Ornstein-Uhlenbeck process, Phys. Rev. E99, 062221 (2019)

    2019

  23. [23]

    A. J. McKane and T. J. Newman, Predator-prey cycles from resonantamplificationofdemographicstochasticity,Phys.Rev. Lett.94, 218102 (2005)

    2005

  24. [24]

    A. J. McKane, J. D. Nagy, T. J. Newman, and M. O. Stefanini, Amplifiedbiochemicaloscillationsincellularsystems,J.Stat. Phys.128, 165 (2007)

    2007

  25. [25]

    P. O. Westermark, D. K. Welsh, H. Okamura, and H. Herzel, Quantificationofcircadianrhythmsinsinglecells,PLOSCom- put. Biol.5, e1000580 (2009)

    2009

  26. [26]

    E.Wallace,M.Benayoun,W.VanDrongelen,andJ.D.Cowan, Emergent oscillations in networks of stochastic spiking neu- rons, PLOS ONE6, e14804 (2011)

    2011

  27. [27]

    Gilson, E

    M. Gilson, E. Tagliazucchi, and R. Cofré, Entropy production of multivariate Ornstein-Uhlenbeck processes correlates with consciousness levels in the human brain, Phys. Rev. E107, 024121 (2023)

    2023

  28. [28]

    Sekizawa, S

    D. Sekizawa, S. Ito, and M. Oizumi, Decomposing thermody- namic dissipation of linear Langevin systems via oscillatory modesanditsapplicationtoneuraldynamics,Phys.Rev.X14, 041003 (2024)

    2024

  29. [29]

    Nartallo-Kaluarachchi, M

    R. Nartallo-Kaluarachchi, M. Kringelbach, G. Deco, R. Lam- biotte, and A. Goriely, Nonequilibrium physics of brain dy- namics, Phys. Rep.1152, 1 (2026)

    2026

  30. [30]

    Dinis, P

    L. Dinis, P. Martin, J. Barral, J. Prost, and J. Joanny, Fluctuation-response theorem for the active noisy oscillator of the hair-cell bundle, Phys. Rev. Lett.109, 160602 (2012)

    2012

  31. [31]

    Alonso, A

    D. Alonso, A. J. McKane, and M. Pascual, Stochastic amplifi- cation in epidemics, J. R. Soc. Interface4, 575 (2007)

    2007

  32. [32]

    R.Kuske,L.F.Gordillo,andP.Greenwood,Sustainedoscilla- tions via coherence resonance in SIR, J. Theor. Biol.245, 459 (2007)

    2007

  33. [33]

    C.A.LugoandA.J.McKane,Quasicyclesinaspatialpredator- prey model, Phys. Rev. E78, 051911 (2008)

    2008

  34. [34]

    Hwang, S.-Y

    C.-R. Hwang, S.-Y. Hwang-Ma, and S.-J. Sheu, Accelerating gaussian diffusions, Ann. Appl. Probab.3, 897 (1993)

    1993

  35. [35]

    T.Lelievre,F.Nier,andG.A.Pavliotis,Optimalnon-reversible lineardriftfortheconvergencetoequilibriumofadiffusion,J. Stat. Phys.152, 237 (2013)

    2013

  36. [36]

    J. B. Weiss, Coordinate invariance in stochastic dynamical systems, Tellus A55, 208 (2003)

    2003

  37. [37]

    Y. V. Fyodorov, E. Gudowska-Nowak, M. A. Nowak, and W. Tarnowski, Nonorthogonal eigenvectors, fluctuation- dissipation relations, and entropy production, Phys. Rev. Lett. 134, 087102 (2025)

    2025

  38. [38]

    Goriely, and R

    R.Nartallo-Kaluarachchi,M.Asllani,G.Deco,M.L.Kringel- bach, A. Goriely, and R. Lambiotte, Broken detailed balance andentropyproductionindirectednetworks,Phys.Rev.E110, 034313 (2024)

    2024

  39. [39]

    D.SornetteandV.Troude,Lifeasnon-normalchemicalaccel- erator, arXiv preprint arXiv:2512.18438 (2025)

    work page arXiv 2025

  40. [40]

    Troude and D

    V. Troude and D. Sornette, Pseudo-coherence and stochastic synchronization: A non-normal route to collective dynamics without oscillators, arXiv preprint arXiv:2603.07206 (2026)

    work page arXiv 2026

  41. [41]

    Sekimoto,Stochastic Energetics(Springer Berlin Heidel- berg, 2010)

    K. Sekimoto,Stochastic Energetics(Springer Berlin Heidel- berg, 2010)

    2010

  42. [42]

    Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep

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

    2012

  43. [43]

    Shiraishi,An Introduction to Stochastic Thermodynamics: From Basic to Advanced(Springer Nature Singapore, 2023)

    N. Shiraishi,An Introduction to Stochastic Thermodynamics: From Basic to Advanced(Springer Nature Singapore, 2023)

    2023

  44. [44]

    Falasco and M

    G. Falasco and M. Esposito, Macroscopic stochastic thermo- dynamics, Rev. Mod. Phys.97, 015002 (2025)

    2025

  45. [45]

    A.C.BaratoandU.Seifert,Coherenceofbiochemicaloscilla- tions is bounded by driving force and network topology, Phys. Rev. E95, 062409 (2017)

    2017

  46. [46]

    Uhl and U

    M. Uhl and U. Seifert, Affinity-dependent bound on the spec- trum of stochastic matrices, J. Phys. A52, 405002 (2019)

    2019

  47. [47]

    N. Ohga, S. Ito, and A. Kolchinsky, Thermodynamic bound on the asymmetry of cross-correlations, Phys. Rev. Lett.131, 077101 (2023)

    2023

  48. [48]

    G.-H. Xu, A. Kolchinsky, J.-C. Delvenne, and S. Ito, Thermo- dynamic geometric constraint on the spectrum of Markov rate matrices, arXiv preprint arXiv:2507.08938 (2025)

    work page arXiv 2025

  49. [49]

    Cycle affinity and winding localize eigenvalues of Markov generators

    A. Kolchinsky, N. Ohga, and S. Ito, Cycle affinity and wind- ing localize eigenvalues of Markov generators, arXiv preprint arXiv:2605.15884 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  50. [50]

    H.QianandM.Qian,Pumpedbiochemicalreactions,nonequi- librium circulation, and stochastic resonance, Phys. Rev. Lett. 84, 2271 (2000)

    2000

  51. [51]

    Nguyen, U

    B. Nguyen, U. Seifert, and A. C. Barato, Phase transition in thermodynamically consistent biochemical oscillators, J. Chem. Phys.149, 045101 (2018)

    2018

  52. [52]

    Oberreiter, U

    L. Oberreiter, U. Seifert, and A. C. Barato, Subharmonic os- cillations in stochastic systems under periodic driving, Phys. Rev. E100, 012135 (2019)

    2019

  53. [53]

    Marsland III, W

    R. Marsland III, W. Cui, and J. M. Horowitz, The thermody- namic uncertainty relation in biochemical oscillations, J. R. Soc. Interface16, 20190098 (2019)

    2019

  54. [54]

    Del Junco and S

    C. Del Junco and S. Vaikuntanathan, Robust oscillations in multi-cyclic Markov state models of biochemical clocks, J. Chem. Phys.152, 055101 (2020)

    2020

  55. [55]

    Del Junco and S

    C. Del Junco and S. Vaikuntanathan, High chemical affinity increasestherobustnessofbiochemicaloscillations,Phys.Rev. E101, 012410 (2020)

    2020

  56. [56]

    Oberreiter, U

    L. Oberreiter, U. Seifert, and A. C. Barato, Stochastic discrete time crystals: Entropy production and subharmonic synchro- nization, Phys. Rev. Lett.126, 020603 (2021)

    2021

  57. [57]

    Remlein, V

    B. Remlein, V. Weissmann, and U. Seifert, Coherence of os- cillations in the weak-noise limit, Phys. Rev. E105, 064101 (2022)

    2022

  58. [58]

    Shiraishi, Entropy production limits all fluctuation oscilla- tions, Phys

    N. Shiraishi, Entropy production limits all fluctuation oscilla- tions, Phys. Rev. E108, L042103 (2023)

    2023

  59. [59]

    Kolchinsky, N

    A. Kolchinsky, N. Ohga, and S. Ito, Thermodynamic bound on spectral perturbations, with applications to oscillations and relaxation dynamics, Phys. Rev. Res.6, 013082 (2024)

    2024

  60. [60]

    Pietzonka and F

    P. Pietzonka and F. Coghi, Thermodynamic cost for precision of general counting observables, Phys. Rev. E109, 064128 (2024)

    2024

  61. [61]

    Gao, H.-M

    Q. Gao, H.-M. Chun, and J. M. Horowitz, Thermodynamic constraints on kinetic perturbations of homogeneous driven diffusions, Europhys. Lett.146, 31001 (2024)

    2024

  62. [62]

    Zheng and E

    C. Zheng and E. Tang, A topological mechanism for robust and efficient global oscillations in biological networks, Nat. Commun.15, 6453 (2024). 7

    2024

  63. [63]

    Hänggi and H

    P. Hänggi and H. Thomas, Stochastic processes: Time evo- lution, symmetries and linear response, Phys. Rep.88, 207 (1982)

    1982

  64. [64]

    Oberreiter, U

    L. Oberreiter, U. Seifert, and A. C. Barato, Universal minimal cost of coherent biochemical oscillations, Phys. Rev. E106, 014106 (2022)

    2022

  65. [65]

    Santolin and G

    D. Santolin and G. Falasco, Dissipation bounds the coherence ofstochasticlimitcycles,Phys.Rev.Lett.135,057101(2025)

    2025

  66. [66]

    Duality between dissipation-coherence trade-off and thermodynamic speed limit based on thermodynamic uncertainty relation for stochastic limit cycles

    R. Nagayama and S. Ito, Duality between dissipation- coherence trade-off and thermodynamic speed limit based on thermodynamicuncertaintyrelationforstochasticlimitcycles, arXiv preprint arXiv:2509.06421 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  67. [67]

    Elementary derivation of the dissipation-coherence bound for stochastic oscillators

    A. Kolchinsky, Elementary derivation of the dissipation– coherence bound for stochastic oscillators, arXiv preprint arXiv:2510.14101 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  68. [68]

    L. N. Trefethen and M. Embree,Spectra and pseudospectra: the behavior of nonnormal matrices and operators(Princeton University Press, 2020)

    2020

  69. [69]

    B.F.FarrellandP.J.Ioannou,Variancemaintainedbystochas- tic forcing of non-normal dynamical systems associated with linearly stable shear flows, Phys. Rev. Lett.72, 1188 (1994)

    1994

  70. [70]

    Penland and P

    C. Penland and P. D. Sardeshmukh, The optimal growth of tropical sea surface temperature anomalies, J. Clim.8, 1999 (1995)

    1999

  71. [71]

    Hennequin, T

    G. Hennequin, T. P. Vogels, and W. Gerstner, Non-normal amplification in random balanced neuronal networks, Phys. Rev. E86, 011909 (2012)

    2012

  72. [72]

    Biancalani, F

    T. Biancalani, F. Jafarpour, and N. Goldenfeld, Giant amplifi- cation of noise in fluctuation-induced pattern formation, Phys. Rev. Lett.118, 018101 (2017)

    2017

  73. [73]

    B. F. Farrell and P. J. Ioannou, Generalized stability theory. Part I: Autonomous operators, J. Atmos. Sci.53, 2025 (1996)

    2025

  74. [74]

    Ahmadian, F

    Y. Ahmadian, F. Fumarola, and K. D. Miller, Properties of networks with partially structured and partially random con- nectivity, Phys. Rev. E91, 012820 (2015)

    2015

  75. [75]

    Polettini and M

    M. Polettini and M. Esposito, Nonconvexity of the relative entropyforMarkovdynamics: AFisherinformationapproach, Phys. Rev. E88, 012112 (2013)

    2013

  76. [76]

    Kondepudi and I

    D. Kondepudi and I. Prigogine,Modern thermodynamics: fromheatenginestodissipativestructures(JohnWiley&Sons, 2014)

    2014

  77. [77]

    D. A. Beard and H. Qian, Relationship between thermody- namicdrivingforceandone-wayfluxesinreversibleprocesses, PLOS ONE2, e144 (2007)

    2007

  78. [78]

    Maes, Local detailed balance, SciPost Phys

    C. Maes, Local detailed balance, SciPost Phys. Lect. Notes , 32 (2021)

    2021

  79. [79]

    G.T.Landi,T.Tomé,andM.J.DeOliveira,Entropyproduction in linear Langevin systems, J. Phys. A46, 395001 (2013)

    2013

  80. [80]

    Godrèche and J.-M

    C. Godrèche and J.-M. Luck, Characterising the nonequilib- rium stationary states of Ornstein–Uhlenbeck processes, J. Phys. A52, 035002 (2018)

    2018

Showing first 80 references.