After Grinstein and Linsker (2007) showed that an error invalidates Dewar's information-theoretic derivation of the maximum entropy production (MaxEP) principle outside the near-equilibrium linear regime, and that Dewar's argument linking MaxEP to self-organized criticality (SOC) is unjustified, the foundational question was left explicitly open: does any form of MaxEP, or a related extremal principle, hold for far-from-equilibrium stationary states with nonlinear constitutive relations, and does MaxEP underlie SOC in slowly driven systems? The subsequent literature (critiques, Dewar's heuristic replies, conditional and model-level results, counterexamples to specific strong versions, and authoritative status reviews through 2021-2023) has not closed either question. No rigorous derivation of any general MaxEP-type principle for the nonlinear far-from-equilibrium regime exists, and no general impossibility result exists; the MaxEP-SOC conjecture remains unproven and unrefuted beyond model-level consistency tests. The problem therefore remains open.

Background

Near equilibrium, irreversible thermodynamics describes a system by thermodynamic forces (affinities) λ\boldsymbol{\lambda} and conjugate fluxes F\boldsymbol{F}, whose relation (the constitutive relation) is linear in the near-equilibrium regime. Entropy production is the product of forces and fluxes summed over transport channels. Two classical extremal statements exist in that linear regime: Prigogine's minimum entropy production principle (a steady state close to equilibrium minimizes entropy production under suitable constraints) and Ziegler's maximum entropy production principle, which selects the actual constitutive relation among all relations compatible with the second law by maximizing dissipation at fixed forces. Neither has an accepted extension to systems far from equilibrium, where the constitutive relations between forces and fluxes are nonlinear.

Jaynes' maximum entropy (MaxEnt) principle is an inference rule: among all probability distributions compatible with given constraints, choose the one maximizing information (Shannon) entropy. Dewar (2003, 2005) claimed to derive a maximum entropy production principle (MaxEP) from MaxEnt applied to trajectories (paths) of nonequilibrium systems, using a fluctuation theorem relating the probability of a flux configuration to that of its time reverse. He claimed the result holds even far from equilibrium, and additionally that self-organized criticality (SOC) - the tendency of slowly driven dissipative systems to settle into a critical state with scale-invariant (power-law) avalanche statistics, paradigmatically in sandpile models - follows necessarily from MaxEP in the limit of infinitely slow driving.

Grinstein and Linsker (2007) showed both claims rest on faulty steps. First, the derivation extends a quadratic (Gaussian) approximation to the flux probability density, valid only near the mean flux, to all fluxes; this implicitly forces the curvature matrix AA to be flux-independent, which is equivalent to linear constitutive relations, and the resulting 'orthogonality condition' needed for the extremal principles fails in the nonlinear regime. In an exactly solvable spin-chain model, the two sides of the claimed constitutive relation λ(F)=A(F)F\lambda(F) = A(F)F evaluate to τ arctanh F\tau\,\mathrm{arctanh}\,F and τF/(1−F2)\tau F/(1-F^2) respectively (with τ\tau the chain length), and agree only in the limit F→0F \to 0 where the constitutive relation becomes linear. Second, the SOC argument computes the flux variance under a mean-field quadratic form assumed valid at all fluxes; retaining the quartic term gives a finite variance, so the claimed divergence (the hallmark of power-law avalanches) is an artifact. The comment closes by stating that the existence of extremal principles (in particular MaxEP) for far-from-equilibrium regimes with nonlinear constitutive relations 'has not been settled', and that Dewar's conjecture that MaxEP or a similar principle underlies SOC, though intuitively appealing, 'has yet to be demonstrated'.

Subsequent work sharpened rather than settled the question: further critiques (Paquette 2011), Dewar's own responses conceding that a rigorous far-from-equilibrium derivation 'remains to be shown' (Dewar 2009) and offering a 'largely heuristic' Liouville-theorem argument (Dewar and Maritan 2011), conditional derivations for restricted classes (Niven 2009), explicit stochastic counterexamples to the beyond-linear-regime claims (Polettini 2013) and to MaxEP as a stability criterion in a nonlinear PDE (Di Vita 2014), model-level supporting tests (Mihelich et al. 2014; Endres 2017; Main and Naylor 2008, 2010 for the SOC case), and authoritative reviews (Martyushev and Seleznev 2006; Martyushev 2013, 2021) that classify the statistical-mechanical foundation of MaxEP as an open problem.

Problem Statement

Determine the status, after the collapse of Dewar's information-theoretic derivation, of maximum-entropy-production-type extremal principles for far-from-equilibrium systems - a question posed explicitly by Grinstein and Linsker (2007) and still open. The problem has two linked components; the second is the slowly driven (self-organized criticality) instance of the first.

(1) Far-from-equilibrium extremal principles. Precisely formulate a candidate maximum entropy production (MaxEP) principle - or a related extremal principle - by specifying the system class, the objective functional (entropy production or dissipation), the constraint set (fixed thermodynamic forces or fluxes, conserved quantities, boundary conditions, ensemble over trajectories or fluxes), and the selection claim (which steady state, phenomenological relation, or transport coefficient is selected). Then either (a) derive that principle rigorously from nonequilibrium statistical mechanics (microscopic dynamics, fluctuation theorems, or large-deviation theory, the mathematical framework characterizing the exponential suppression of the probabilities of atypical fluctuations such as rare currents) for a class of stationary states defined by structural conditions (rather than a single model or a single parameter family) whose constitutive relations are genuinely nonlinear, i.e. far from equilibrium; or (b) prove a no-go theorem showing that no extremal principle of the stated kind holds across the far-from-equilibrium nonlinear regime as a whole, and not merely within a proper subclass of it - where, since the question asks whether any form of MaxEP or a related extremal principle holds, a decisive no-go must quantify over a well-defined space of candidate extremal principles (including the stated formulation); an explicit counterexample refuting only one specific formulation, or a no-go confined to a proper subclass of that regime, is partial progress toward such a no-go, not a complete negative answer.

(2) The MaxEP-SOC conjecture. Establish or refute, by general theoretical argument rather than single-model consistency tests, the conjecture that slowly driven dissipative systems self-organize to critical states with scale-invariant avalanche statistics because such states maximize entropy production (in the limit of infinitely slow driving relative to internal relaxation).

A complete answer must state explicitly which formulation of MaxEP it addresses and why that formulation is faithful to the question (some form of maximum entropy production or a related extremal principle), and must cover both components - a result on component (1) addresses component (2) only if, by general theoretical argument, it establishes or refutes the MaxEP-SOC conjecture itself for slowly driven systems exhibiting self-organized criticality. Answers confined to the near-equilibrium linear regime, or purely heuristic arguments, or conditional derivations for restricted classes already in the literature (e.g. Niven 2009), or single-model consistency studies without a general statement, resolve neither component: the near-equilibrium case (Ziegler's principle, Prigogine's minimum entropy production) is already classical, and model-level supportive evidence (e.g. from earthquake dynamics or stochastic lattice exclusion-process models of particle transport), model-level counterexamples to one specific MaxEP variant, or conditional derivations for restricted classes already exist in the literature without settling the general question.

References

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