Source
Grinstein & Linsker (2007), J. Phys. A: Math. Theor. 40:9717–9720, LIT-525, read in full in NOTE-422 from the authors’ submitted manuscript (the published Comment was not seen). Dewar (2003), LIT-521, read in full in NOTE-426. Dewar’s 2005 letter (J. Phys. A 38:L371), the target of the Comment’s first point, could not be read and is not filed.
What was actually shown
The 2003 route (LIT-521, §4.2). Maximum path entropy gives maximum mean entropy production only within a mean-field approximation that ignores fluctuations of the action (Eq. 23), and only if the number of paths with irreversible action A^irr increases with A^irr (p. 14). The second is assumed with no argument or model where it can be checked (NOTE-426).
The 2005 route (LIT-525, §1). A fluctuation theorem compares p(f) with p(−f). A Gaussian approximation around the mean flux F describes p(f) only near F. Both can hold at once only when f and −f are near F, so F near zero. Dewar’s Eq. 15, λ_k = Σ_j A_jk(F) F_j, needs his quadratic Eq. 14 for all f. Differentiating gives ∂A_jk/∂F_n = 0 (Eq. 2): A is independent of F, which is linearity, “contrary to Dewar’s claim”. The correct gradient of the mean dissipation is ∂D/∂F_n = 2λ_n + 2Σ_k A_nk(F) F_k (Eq. 3, re-derived in NOTE-422). Dewar’s orthogonality condition, ∂D/∂F_n = 4λ_n, holds only when Σ_k A_nk F_k = λ_n, in particular when A is constant.
Both principles fall together. Dewar uses the orthogonality condition to derive “maximum dissipation” and “minimum dissipation”. Both derivations therefore “do require ‘the usual near-equilibrium assumption of linear constitutive relations’” (p. 3), the inner phrase being Dewar’s own for what he claimed to avoid.
The spin chain (§1.1). Paths are τ spins of ±1/τ. Exactly, F = tanh(λ/τ), λ(F) = τ artanh F and A(F) = τ/(1 − F²). The fluctuation theorem holds exactly, and the quadratic expansion matches Dewar’s Eq. 14 near F. But his Eq. 15 would need τ artanh F = τF/(1 − F²), which holds only as F → 0. NOTE-422 checked each step.
Self-organized criticality (§2). In Dewar 2003 the variance of the output flux, ⟨(F − ⟨F⟩)²⟩ ≈ 1/(8|g|F_ext²) (Eq. 28), diverges as the drive F_ext → 0. It comes from expanding H = rF² + gF⁴ to quadratic order about ⟨F⟩ and using that form for all F. The quadratic coefficient vanishes like |g|F_ext². Keeping the quartic term gives ⟨F²⟩ ∝ 1/√|g|, finite at F_ext = 0. NOTE-422 checked this against Dewar 2003, Eqs. 26–28.
What could have come out otherwise. In the spin chain the exact λ(F) could have satisfied Dewar’s Eq. 15 at large F. It does not. With the quartic term kept, the variance could still have diverged. It does not.
What this does not say
- That MEP is false. The Comment removes one derivation and one application, and offers no counterexample to MEP as an empirical principle. It says “the question of the existence of possible extremal principles … that might apply to far-from-equilibrium regimes … has not been settled”.
- That MEP and self-organized criticality are unrelated. It calls the link “an intuitively appealing notion”. Whether a MaxEP model shows self-organized criticality is set by the assumed large-F tail of p(F), of which “no conclusion … can be drawn” in the model as given (p. 5).
- That Dewar 2005 was checked here. The Comment’s account of that letter’s equations is taken as given. The failure is shown on one scalar model.
- That the minimum principle is false. Only that, derived through the orthogonality condition, it needs linear constitutive relations. That is consistent with treating it as a near-equilibrium result.
- That Dewar’s fluctuation theorem is wrong. It is exact given his inferred path measure. What it does not show is that real path statistics follow that measure (NOTE-426).
Connections
- The account it bears on is maximum entropy production (THEORY-671). This finding is why that account cannot be promoted on the strength of its derivation. Kleidon’s review (LIT-519) cites the Comment only for its point about self-organized criticality (NOTE-423).
- Prigogine. The minimum principle’s own statement restricts it to the strictly linear regime (LIT-516), which is where this finding leaves Dewar’s derivation of both principles.
- The dynamical fluctuation theorems (THEORY-431) do not depend on any of this. They are derived from microscopic reversibility, not from maximum-entropy inference.