Source
Rao and Esposito (2016), LIT-1889, §§II C–E and III D, eqs. 20, 28, 50, 55 and 59–61, Example 6, pp. 4–10 of arXiv 1602.07257v3; read in NOTE-1464. The proof that a single chemostat leaves the affinity conservative is Dal Cengio, Lecomte and Polettini (2023), LIT-1897, eq. 47 (NOTE-1476).
What was actually shown
Bookkeeping. Close a network and it has conservation laws ℓ (left null vectors of the stoichiometric matrix ∇) and cycles (right null vectors). Fix the concentrations of a set S_y of species and the internal matrix ∇_X loses rows. By rank–nullity, |S_y| = |λ_b| + |ε|: each chemostatted species either breaks a conservation law or creates an emergent cycle, a set of reactions that leaves the internal species unchanged and moves chemicals from one chemostat to another (pp. 4–5). The first chemostat always breaks total mass, so one chemostat creates no emergent cycle and the network is “unconditionally detailed-balanced” (Example 6).
Equilibrium. Local detailed balance, k⁺/k⁻ = exp(−Δ_r G°/RT), makes every cycle affinity of the closed network zero. An emergent cycle’s affinity A_ε depends only on the chemostatted concentrations (eq. 28). The network is detailed-balanced iff every A_ε = 0, and then relaxes to an equilibrium that is globally stable and unique for given unbroken conserved quantities (pp. 5–6).
Cost. Entropy production is RT Σ_ρ (J⁺ − J⁻) ln(J⁺/J⁻) ≥ 0 (eq. 55); at a nonequilibrium steady state it equals Σ_ε J_ε A_ε (eq. 61).
A worked check (mine). CASE-189’s first construction, two capacities producing each other with saturating production and turnover, is written with irreversible Hill terms. Rewritten as elementary reversible chemistry, with fuel S and waste W chemostatted, S + 2c₂ ⇌ c₁ + 2c₂, S + 2c₁ ⇌ c₂ + 2c₁, c_i ⇌ W, all rate constants 1 and [S] = 3, the network has two chemostats, one broken law, one emergent cycle (S → W) and deficiency 2. Its emergent affinity is RT ln(3/[W]). At [W] = 3, zero affinity, there is one fixed point, c = (3, 3), with no current. The extinct state and the threshold appear only for [W] < 0.0887, above 3.52 RT. As [W] → 0 the symmetric equilibria tend to the case’s own, (3 ∓ √5)/2. At [W] = 0.01 the maintained state dissipates 29.77 (RT times the rate unit), and the reaction-by-reaction sum and J_ε A_ε agree.
What could have failed. The steady dissipation could have depended on internal concentrations in a way not captured by the emergent cycles. Or a single reservoir could have sustained a current. Neither happens, and both are exact.
What this does not say
- That a driven network is organized, multistable or far from equilibrium. A nonzero affinity is necessary. What the network does with it is kinetics. Complex-balanced networks, including every deficiency-zero network, have a unique globally attracting steady state however hard they are driven (LIT-1889, §IV), so they form no dissipative structure.
- That a positive production rate in a phenomenological model is a mechanism. Hill and Michaelis–Menten terms, and irreversible decay, are effective rate laws outside the theory until resolved into elementary steps. The infinite-affinity limit in which they are recovered is where a model’s “no basal production” assumption lives (NOTE-1464, on CLAIM-516).
- Anything about catalytic organization. Catalysts cancel from the stoichiometric matrix, so who produces whom does not enter the thermodynamic budget (THEORY-1248).
- That the chemostats are external. They are a regime of a larger closed system, valid for a window (THEORY-1250).
- Anything beyond ideal dilute, well-mixed, deterministic chemistry.