Ziegler's principle of thermodynamic orthogonality — equivalently, the maximum entropy production principle (MaxEP) applied to constitutive laws — reproduces Onsager's reciprocal relations in the linear-response regime by selecting the symmetric response matrix that maximizes entropy production at fixed external forces. Polettini (2013) showed that, extended beyond linear response, the same principle imposes an infinite hierarchy of higher-order reciprocal relations on the nonlinear response tensors, and that a simple three-state Markov jump process already violates them; the paper therefore questioned whether MaxEP has any nontrivial domain of validity beyond linear response. The open questions it left are: (i) whether any nontrivial physical system satisfies the higher-order reciprocal-relations hierarchy with positive entropy production, or whether an impossibility result confines Ziegler's principle to linear response and trivial special cases; (ii) whether a microscopic dynamics can realize MaxEP by letting constitutive coefficients themselves relax at fixed external forces; (iii) whether system boundaries and coarse-grainings can be systematically chosen so that local entropy production is maximized at the physical steady state; together with two structural bridges: (iv) the general mapping between network and phenomenological nonequilibrium thermodynamics, and (v) whether path-entropy inference (Maximum Caliber, Dewar-type arguments) can ground entropy-production extremal principles. Work since 2013 leaves all five strands open: adjacent programs — Hessian-geometry entropy-production decompositions, steepest-entropy-ascent dynamics, Maximum Caliber inference, experimental self-organized structures — reformulate or bypass the question without settling it.

Background

Near equilibrium, irreversible thermodynamics describes a system by finitely many conjugate pairs of thermodynamic forces FiF^i and fluxes (currents) JiJ_i — for example, a temperature gradient paired with a heat current, or an electrochemical potential difference paired with a particle current. The entropy production rate is σ=∑iFiJi≥0\sigma = \sum_i F^i J_i \ge 0, and the near-equilibrium constitutive law is linear, Fi=∑jMijJjF^i = \sum_j M^{ij} J_j. Onsager's reciprocal relations state that (up to magnetic-field-type terms) the response matrix is symmetric, Mij=MjiM^{ij} = M^{ji}, a fact derivable from microscopic time-reversal invariance. The minimum entropy production principle (MinEP) complements this: for fixed external forces and fixed linear coefficients, the steady state minimizes σ\sigma.

Ziegler proposed a variational counterpart. Assuming that "all power is dissipated", i.e. that there exists a dissipation function ω(J)\omega(J) (quadratic near equilibrium) with σ=ω\sigma = \omega on the constitutive manifold, Ziegler's principle of thermodynamic orthogonality requires the force to be parallel to the gradient of the entropy production with respect to the currents, Fa=λ(J) δσ/δJaF^a = \lambda(J)\, \delta\sigma/\delta J_a. Equivalent to it, the maximum entropy production principle (MaxEP) selects, among all response matrices compatible with the constraint σ≡ω\sigma \equiv \omega at fixed external forces, the one with maximal entropy production; Polettini (2013) proved that in the linear regime this selects exactly the symmetric (Onsager) matrix, so Ziegler's principle reproduces Onsager reciprocity as a selection rule on constitutive laws.

Beyond linear response, writing the forces as analytic functions of the currents, Fa=∑r≥2Lai2…irJi2⋯JirF^a = \sum_{r \ge 2} L^{a i_2 \ldots i_r} J_{i_2} \cdots J_{i_r}, with a compatible positive entropy production σ=∑r≥2Mi1…irJi1⋯Jir\sigma = \sum_{r \ge 2} M^{i_1 \ldots i_r} J_{i_1} \cdots J_{i_r}, orthogonality is equivalent to an infinite, order-by-order family of constraints Tab i1…is=0T^{ab\,i_1 \ldots i_s} = 0 (Equation (23) of the source and its higher-order successors) on the higher-order response tensors; at second order these reduce to Onsager's relations. Polettini then tested the hierarchy in stochastic thermodynamics — the framework in which small nonequilibrium systems are modeled as Markov jump processes on a graph, with edge currents and edge thermodynamic forces (logarithmic ratios of transition rates) organized by Schnakenberg's network theory — and exhibited a three-state Markov jump process with uniform transition rates whose third-order response coefficients violate the hierarchy. The violation is robust: entropy-production-preserving linear reparametrizations of the conjugate variables cannot mix the symmetric and antisymmetric tensor parts, so no choice of macroscopic conjugates repairs it, and slight generalizations of the model also refute the fallback hypothesis that the response tensors are fully symmetric in all indices.

The division of labor between MaxEP and MinEP is as follows. On networks with fixed microscopic linear response, imposing Kirchhoff's current law (conservation of the transported quantity at the network nodes), MaxEP prescribes reciprocal relations between coarse-grained (cycle) observables but does not produce the steady state: the onset of the steady state follows MinEP, in agreement with Polettini's earlier theorem for Markovian networks, and it corrects earlier claims that Kirchhoff's loop law follows from MaxEP. A dynamical MaxEP would instead require a framework in which the response coefficients themselves relax at fixed external forces, so that initially violated Onsager relations are restored in the long-time limit. Polettini also noted that nonequilibrium forces may depend on additional time-reversal-even variables KK beyond the currents, Fa=Fa(J,K)F^a = F^a(J, K), breaking the hypothesis that all power is dissipated; that the system-boundary problem — for which choice of boundaries is a local entropy production maximized at steady state? — had resisted the author's own attempts in linear network and stochastic thermodynamics; and that inference-based methods — Maximum Caliber (maximization of path entropy under dynamical constraints, the dynamical analogue of Jaynes maximum entropy) and Dewar's information-theoretic derivation of MaxEP — had not been reconciled with entropy-production extremization: the Markovian path measure maximizes path entropy, but path entropy "has nothing to do with" entropy production. These five issues — hierarchy existence, dynamical realization, boundary selection, the network/phenomenological mapping, and the inference bridge — remain open.

Problem Statement

Decide the domain of validity of Ziegler's maximum entropy production principle — equivalently, thermodynamic orthogonality — beyond the linear response regime, and identify the mechanisms that could make it hold or prove that none exist. In the framework where thermodynamic forces are analytic functions of the conjugate currents, Fa=∑r≥2Lai2…irJi2⋯JirF^a = \sum_{r \ge 2} L^{a i_2 \ldots i_r} J_{i_2} \cdots J_{i_r}, and the entropy production σ=∑r≥2Mi1…irJi1⋯Jir\sigma = \sum_{r \ge 2} M^{i_1 \ldots i_r} J_{i_1} \cdots J_{i_r} is positive, the problem decomposes into five source-stated components. Here thermodynamic orthogonality is the requirement that there exist a scalar function λ(J)\lambda(J) such that Fa=λ(J) δσ/δJaF^a = \lambda(J)\, \delta\sigma/\delta J_a holds for every index aa; imposed order-by-order in the currents, it is equivalent to an infinite hierarchy of constraints Tab i1…is=0T^{ab\,i_1 \ldots i_s} = 0 on the response tensors (Equation (23) of the source and its higher-order successors), whose second-order sector reproduces Onsager's reciprocal relations:

(1) Constitutive existence and classification. Apart from the trivial special cases in which the entropy production is a homogeneous polynomial of degree rr in the currents or only one force-current pair is considered, do any nontrivial physical systems satisfy the infinite hierarchy of higher-order reciprocal relations (Equation (23) of the source and its successors) that thermodynamic orthogonality imposes on the response tensors, with strictly positive entropy production? If such systems exist, characterize how common or rare they are within physically realizable dynamics, whose admissible class must be fixed and justified explicitly in the answer (canonically, Markov jump processes with local detailed balance, meaning edge thermodynamic forces equal logarithmic ratios of forward and backward transition rates); if they do not, prove an impossibility result confining Ziegler's principle to the linear regime and the trivial special cases.

(2) Dynamical realization. Identify microscopic dynamical laws under which the constitutive (response) coefficients themselves relax in time at fixed external macroscopic forces — so that Onsager reciprocity, violated at some initial time, is restored in the long-time limit and MaxEP, rather than MinEP, is the governing variational principle — or show that no physically consistent (e.g., Markovian) realization of this scenario exists.

(3) System-boundary problem. Determine whether one can systematically identify system boundaries and effective coarse-grainings of a nonequilibrium network for which the resulting local entropy production is maximized at the physical steady state, and formulate the precise criteria that such boundaries must satisfy; or show that no such criteria exist.

(4) Network-phenomenological bridge. Assess whether phenomenological (multi-species) nonequilibrium thermodynamics can always be cast in the form of network (single-conserved-species, Schnakenberg-type) thermodynamics, and whether phenomenological MinEP/MaxEP descend from network variational principles in that generality.

(5) Inference bridge. Assess whether an inference-based maximization of path entropy (Maximum Caliber, Dewar-type arguments) can yield reliable, physically interpretable MaxEP statements about entropy production, or whether the two frameworks are fundamentally distinct.

Components (1)-(3) are the load-bearing core: jointly they decide whether Ziegler's MaxEP is a genuine physical principle beyond linear response or an artifact of that regime. Components (4)-(5) are the structural bridges the source explicitly flags as unresolved, which would supply derivations or delimitations of the core. A complete answer must resolve each of the core components (1)-(3) — by existence results, classifications, or impossibility theorems — and, for each of the bridges (4)-(5), need only state whether and how it bears on the verdict; the bridges are not independently required deliverables.

References

  1. Matteo Polettini, 'Fact-Checking Ziegler's Maximum Entropy Production Principle beyond the Linear Regime and towards Steady States', Entropy 15(7), 2570 (2013). DOI 10.3390/e15072570, https://doi.org/10.3390/e15072570 (preprint arXiv:1307.2052, https://arxiv.org/abs/1307.2052).
  2. H. Ziegler, 'An Introduction to Thermomechanics', North-Holland, Amsterdam (1983). Book DOI 10.1016/c2009-0-07441-8, https://doi.org/10.1016/c2009-0-07441-8; chapter 'Thermodynamic Orthogonality', DOI 10.1016/b978-0-444-86503-8.50018-4, https://doi.org/10.1016/b978-0-444-86503-8.50018-4 (cited as in the source's reference list).
  3. Lars Onsager, 'Reciprocal Relations in Irreversible Processes. I.', Physical Review 37, 405 (1931). DOI 10.1103/PhysRev.37.405, https://doi.org/10.1103/PhysRev.37.405
  4. Matteo Polettini, 'Macroscopic constraints for the minimum entropy production principle', Physical Review E 84, 051117 (2011). DOI 10.1103/PhysRevE.84.051117, https://doi.org/10.1103/PhysRevE.84.051117
  5. Paško Županović, Davor Juretić, and Srećko Botrić, 'Kirchhoff's loop law and the maximum entropy production principle', Physical Review E 70, 056108 (2004). DOI 10.1103/PhysRevE.70.056108, https://doi.org/10.1103/PhysRevE.70.056108
  6. James Hurley and Claude Garrod, 'Generalization of the Onsager Reciprocity Theorem', Physical Review Letters 48, 1575 (1982). DOI 10.1103/PhysRevLett.48.1575, https://doi.org/10.1103/PhysRevLett.48.1575
  7. P. P. Barashev, 'Onsager reciprocity relations for a nonlinear functional dependence of the thermodynamic forces and fluxes', Soviet Physics Journal 12, 1496 (1969). DOI 10.1007/BF00838313, https://doi.org/10.1007/BF00838313
  8. David Andrieux and Pierre Gaspard, 'A fluctuation theorem for currents and non-linear response coefficients', Journal of Statistical Mechanics: Theory and Experiment (2007) P02006. DOI 10.1088/1742-5468/2007/02/P02006, https://doi.org/10.1088/1742-5468/2007/02/P02006
  9. David Andrieux and Pierre Gaspard, 'Fluctuation Theorem for Currents and Schnakenberg Network Theory', Journal of Statistical Physics 127, 107 (2007). DOI 10.1007/s10955-006-9233-5, https://doi.org/10.1007/s10955-006-9233-5
  10. Nathaniel Virgo, 'From Maximum Entropy to Maximum Entropy Production: A New Approach', Entropy 12(1), 107 (2010). DOI 10.3390/e12010107, https://doi.org/10.3390/e12010107
  11. V. D. Seleznev and L. M. Martyushev, 'Fluctuations, Trajectory Entropy, and Ziegler's Maximum Entropy Production', arXiv:1112.2848, https://arxiv.org/abs/1112.2848 (published version in 'Beyond the Second Law: Entropy Production and Non-equilibrium Systems', Springer, 2014)
  12. R. C. Dewar, 'Maximum entropy production and the fluctuation theorem', Journal of Physics A: Mathematical and General 38, L371 (2005). DOI 10.1088/0305-4470/38/21/L01, https://doi.org/10.1088/0305-4470/38/21/L01
  13. G. Grinstein and R. Linsker, 'Comments on a derivation and application of the "maximum entropy production" principle', Journal of Physics A: Mathematical and Theoretical 40, N01 (2007). DOI 10.1088/1751-8113/40/31/N01, https://doi.org/10.1088/1751-8113/40/31/N01
  14. Stijn Bruers, 'A discussion on maximum entropy production and information theory', Journal of Physics A: Mathematical and Theoretical 40, 7441 (2007). DOI 10.1088/1751-8113/40/27/003, https://doi.org/10.1088/1751-8113/40/27/003
  15. R. C. Dewar and A. Maritan, 'A Theoretical Basis for Maximum Entropy Production', in 'Beyond the Second Law: Entropy Production and Non-equilibrium Systems', Springer (2014), pp. 49-71. DOI 10.1007/978-3-642-40154-1_3, https://doi.org/10.1007/978-3-642-40154-1_3
  16. A. Mielke, D. R. M. Renger, and M. A. Peletier, 'A Generalization of Onsager's Reciprocity Relations to Gradient Flows with Nonlinear Mobility', Journal of Non-Equilibrium Thermodynamics 41, 141 (2016). DOI 10.1515/jnet-2015-0073, https://doi.org/10.1515/jnet-2015-0073 (preprint arXiv:1510.06219, https://arxiv.org/abs/1510.06219)
  17. Francesco Benfenati and Gian Paolo Beretta, 'Ergodicity, Maximum Entropy Production, and Steepest Entropy Ascent in the Proofs of Onsager's Reciprocal Relations', Journal of Non-Equilibrium Thermodynamics 43, 101 (2018). DOI 10.1515/JNET-2017-0054, https://doi.org/10.1515/JNET-2017-0054
  18. A. Belkin, A. Hubler, and A. Bezryadin, 'Self-Assembled Wiggling Nano-Structures and the Principle of Maximum Entropy Production', Scientific Reports 5, 8323 (2015). DOI 10.1038/srep08323, https://doi.org/10.1038/srep08323
  19. Alexey Bezryadin and Erik D. Kountz, 'Training Concept, Evolution Time, and the Maximum Entropy Production Principle', Entropy 18(4), 145 (2016). DOI 10.3390/e18040145, https://doi.org/10.3390/e18040145
  20. Albert Han, Jiri Kataman-Kustwan, and Alexey Bezryadin, 'Competition for Survival and the Maximum Entropy Production Principle in Self-Organized Silver Particle Chains', arXiv:2605.27457 (2026), https://arxiv.org/abs/2605.27457
  21. James Glimm, Daniel Lazarev, and Gui-Qiang Chen, 'Maximum entropy production as a necessary admissibility condition for the fluid Navier-Stokes and Euler equations', arXiv:2003.05517 (2020), https://arxiv.org/abs/2003.05517
  22. Rohit Kishan Ray, 'Finite-Dimensional Quantum Systems under the Fourth Law of Thermodynamics', arXiv:2508.14389 (2025), https://arxiv.org/abs/2508.14389
  23. Matteo Polettini and Massimiliano Esposito, 'Effective Fluctuation and Response Theory', Journal of Statistical Physics 176, 94 (2019). DOI 10.1007/s10955-019-02291-7, https://doi.org/10.1007/s10955-019-02291-7 (preprint arXiv:1803.03552, https://arxiv.org/abs/1803.03552)
  24. Tetsuya J. Kobayashi, Dimitri Loutchko, Atsushi Kamimura, and Yuki Sughiyama, 'Hessian geometry of nonequilibrium chemical reaction networks and entropy production decompositions', Physical Review Research 4, 033208 (2022). DOI 10.1103/PhysRevResearch.4.033208, https://doi.org/10.1103/PhysRevResearch.4.033208 (preprint arXiv:2206.00863, https://arxiv.org/abs/2206.00863, titled 'Geometry of Nonequilibrium Chemical Reaction Networks and Generalized Entropy Production Decompositions')
  25. Purushottam D. Dixit, Jason Wagoner, Corey Weistuch, Steve Pressé, Kingshuk Ghosh, and Ken A. Dill, 'Perspective: Maximum caliber is a general variational principle for dynamical systems', The Journal of Chemical Physics 148 (2018). DOI 10.1063/1.5012990, https://doi.org/10.1063/1.5012990 (preprint arXiv:1711.03450, https://arxiv.org/abs/1711.03450)
  26. Kingshuk Ghosh, Purushottam D. Dixit, Luca Agozzino, and Ken A. Dill, 'The Maximum Caliber Variational Principle for Nonequilibria', Annual Review of Physical Chemistry 71, 213 (2020). DOI 10.1146/annurev-physchem-071119-040206, https://doi.org/10.1146/annurev-physchem-071119-040206
  27. L. M. Martyushev, 'Maximum entropy production principle: history and current status', Physics-Uspekhi 64, 558 (2021). DOI 10.3367/UFNe.2020.08.038819, https://doi.org/10.3367/UFNe.2020.08.038819
  28. J. Svoboda, K. Hackl, and F. D. Fischer, 'A note on maxima and minima of dissipation in context of treatment of irreversible thermodynamic systems by application of Extremal Principles', Scripta Materialia 233, 115519 (2023). DOI 10.1016/j.scriptamat.2023.115519, https://doi.org/10.1016/j.scriptamat.2023.115519
  29. F. D. Fischer, G. A. Zickler, K. Hackl, and J. Svoboda, 'A note concerning some aspects for application of a thermodynamic extremal principle (TEP) for continua', Acta Mechanica 235, 3309 (2024). DOI 10.1007/s00707-024-03892-4, https://doi.org/10.1007/s00707-024-03892-4
  30. P. A. Tadjibaev and D. P. Tadjibaev, 'Extremums of Entropy Production and Second Law in Rayleigh Gas', arXiv:1610.00499 (2016), https://arxiv.org/abs/1610.00499