Source
Vellela & Qian (2009), LIT-1890, §3.3 (eqs. 3.12–3.27, Figs. 6–7) and §5, read in NOTE-1474. Falasco & Esposito (2025), LIT-1892, §III.F (eqs. 101–102), §IV.H (eq. 160) and §VIII.B.6 (eqs. 290–292), read in NOTE-1472.
What was actually shown
The counterexample. Each deterministic steady state xᵢ* of the Schlögl model dissipates at ξᵢ = ΔG·J^ss_i ≥ 0 (eqs. 3.12–3.14). The master equation has one mean entropy production rate, ξ_CME = ΔG × (stationary cycle flux) (eq. 3.23), approximately V times the ξᵢ of the two stable states weighted by their probabilities (eq. 3.27). Sweeping b at a = 1 (Fig. 6), ξ_CME/V jumps from the lower branch ξ₋ to the upper ξ₊ where the dominant state changes. So the state the system occupies is sometimes the more and sometimes the less dissipative of the two: “the more stable of the two stable equilibria can have either a high or a low entropy production rate” (§3.3); “the entropy production rate alone is unable to fully describe stability in the steady state” (§5). The middle state’s rate, always between the other two, is never represented, so ξ_CME is always the maximum or the minimum of the three.
The general account. Minimum entropy production is recovered for states close to detailed balance under linearized dynamics (Falasco & Esposito eqs. 101–102), “only” there. Read as a statement about lifetimes, eq. (160) is a maximum principle, “the attractor with the largest life-time is the one with the largest relaxation (or smallest instantonic) entropy production”. It is exact at linear order and tight only where the two path entropies nearly coincide, which in general requires ⟨σ̇_γ⟩ → 0. The usual statement, that the attractor with the largest steady dissipation lives longest, needs that dissipation to be concentrated near the fixed points and the relaxation and instanton time scales to match. “Some fine tuning or peculiar symmetries” (eqs. 290–292).
What could have failed. The stochastic rate might have been pinned to one branch, or the more probable branch might always have been the higher (or lower) one. In the sweep it is not.
What this does not say
- That dissipation is irrelevant to stability. The escape rate is bracketed by path entropies (THEORY-1264); dissipation along trajectories constrains lifetimes. What fails is ranking states by their steady dissipation rate alone.
- That Fig. 7’s trend is a principle. Vellela & Qian’s observation that entropy production falls as the dominant state’s barrier rises, offered as support for least dissipation as robustness, is one sweep of one parameter in one model.
- That every maximum-entropy-production claim is refuted. THEORY-671 states a selection principle for systems with many internal degrees of freedom. This is a one-variable counterexample plus a general argument that the principle needs special conditions; it does not show that the conditions never hold in the systems THEORY-671 is about.