Source
Vellela & Qian (2009), LIT-1890, §§2–4 and appendix A, read in NOTE-1474 from the PMC copy. Falasco & Esposito (2025), LIT-1892, §III.C (multistability against metastability), §IV.H (eq. 157) and §VI, read in NOTE-1472.
What was actually shown
The model. A + 2X ⇌ 3X, B ⇌ X with A and B chemostatted, in volume V. The rate equation ẋ = k₁ax² − k₂x³ − k₄x + k₃b has two stable states and a saddle in a region of (a, b), which requires the chemostats out of equilibrium, k₁k₄a ≠ k₂k₃b (Vellela & Qian eqs. 2.3, 3.6, 3.10). The chemical master equation is a birth–death chain with a unique stationary distribution p^ss_n ∝ Π λᵢ/μᵢ₊₁ (eq. 2.8), two-peaked where the rate equation is bistable.
The time scales. The generator’s second eigenvalue λ₁ is exponentially small in V and equals −(r₊ + r₋), the sum of the escape rates of a two-state reduction (§§3.4, 4.1); the other eigenvalues stay of order one (Table 1: −1.64, −2.37, −3.18 at V = 100). Asymptotically ln rᵢ ≈ −VΔuᵢ, with Δu the drop in the quasi-potential u(x) = ∫ ln(J⁺/J⁻) from the stable state to the saddle (eqs. 3.29–3.30). Checked on eq. (4.4)‘s parameters: λ₁ from the full generator matches a Kramers-type sum to four digits for V = 40–80, and ln|λ₁| falls at −0.22 per unit V, the smaller of the two barriers (0.222 and 0.318).
Initial-condition independence. The long-run share of time near each state is P₋/P₊ = r₊/r₋, set by u(x₋) − u(x₊) at leading order. For large V the system “will almost surely be found in only one of two FCA. Which state it ends up in depends solely on the parameter values … and not on the initial condition!” (§3.4). In Falasco & Esposito’s terms the V → ∞ and t → ∞ limits do not commute; the deterministic limit is multistable while the mesoscopic law is unique (§III.C).
Finite-size exchange of stability. At a = 1, b = 2 the favoured state moves from x₋ to x₊ as V grows (Fig. 4). Checked: the lower well holds 73%, 38% and 2% of the mass at V = 10, 20, 40. The O(1) prefactor of the asymptotic ratio places the crossover at V ≈ 16.5.
The fold (my check, not in either paper). As b approaches the saddle-node at b* = 3.278 (a = 1), the lower state’s barrier closes as (b* − b)^{3/2}: barrier/(b* − b)^{3/2} = 0.064 for b* − b from 0.1 to 0.001. So the lifetime e^{VΔu} collapses before the deterministic fold.
What could have failed. The slow eigenvalue might not have separated from the rest, or the stationary weights might have depended on initial data through a conserved quantity. Neither happens; the chain is irreducible.
What this does not say
- That the more stable state is the deeper well of any energy. The barrier is in a quasi-potential computed from the rates. Its sign is not given by the rate equation’s “potential” ∫ẋ dx (Vellela & Qian footnote 1), nor reliably by a Fokker–Planck approximation (THEORY-1265).
- That the paper’s asymptotic rates are right. As printed, eqs. (3.29)–(3.30) carry a factor 1/V that puts them below the exact λ₁ by a factor growing like V (0.59V at V = 150). The paper’s reported “breakdown” of asymptotics at small volume is mostly that factor (NOTE-1474).
- That lifetime fixes stationary probability in general. With two states it does, P₋/P₊ = τ₋/τ₊. With three or more attractors the routing between them matters, and Falasco & Esposito note that a long-lived dissipative state can be improbable (§VIII.B.6, citing Maes & Netočný, not read).
- Anything about two or more species. The closed-form stationary law is a one-dimensional accident; in higher dimension the quasi-potential must be found from a Hamilton–Jacobi equation (LIT-1892, eq. 34).
- That deterministic constructions such as CASE-189 or CASE-111 are wrong. Their folds are the V → ∞ statements. What changes at finite size is that a maintained regime has a lifetime, and loses it before the fold.