Source
Santolin et al. (2025), LIT-1881, §§IV–VIII (eqs. 32, 45, 61–76), read in full in NOTE-1465. Falasco & Esposito (2025), LIT-1892, §IV.B (eq. 119), §IV.H (eqs. 157–166), §VII.A.1 and appendix A, read in NOTE-1472.
What was actually shown
The rate. With P_ss ≍ e^{−ΩI_ss} for a large parameter Ω (a volume, an inverse noise strength), the escape rate from stable point xᵢ through saddle x_ν is r ≍ e^{−Ω[I_ss(x_ν) − I_ss(xᵢ)]} (Santolin eq. 63; Falasco & Esposito eq. 157). This is Freidlin–Wentzell theory. I_ss is the quasi-potential, a Lyapunov function of the noiseless dynamics, equal to the free energy (Massieu potential) only at detailed balance.
The bracket. Write the drift as u = −d·∇I_ss + v with v·∇I_ss = 0 (Santolin eq. 45; Falasco & Esposito eq. 71). Relaxation follows ẋ = v − d∇I_ss and the instanton ẋ = v + d∇I_ss (eq. 61). The entropy flow along either is ∓ΔI_ss plus σ_a = ∫ v·d⁻¹·v ≥ 0 (eqs. 68–70). Hence σ_{ν→i} = σ_a + ΔI_ss ≥ ΔI_ss and σ_{i→ν} = σ_a − ΔI_ss ≥ −ΔI_ss (eqs. 72–73), and −σ_{ν→i} ≤ lim Ω⁻¹ ln r ≤ σ_{i→ν} (eq. 76). For jump processes the same bracket follows from the emergent second law and a Jensen argument for the adiabatic part along instantons (Falasco & Esposito eq. 160, appendix A). At detailed balance v = 0, the instanton is the reversed relaxation, and the bounds meet. To first order in the nonconservative force, I_ss = ϕ plus that force’s dissipation along the detailed-balanced relaxation, and the bracket is again an equality (Falasco & Esposito eq. 120; Santolin eq. 38).
A check. On dx = −Ax dt + √2 dW with A = [[1, g], [−g, 2]] (closed-form I_ss), both inequalities hold for g = 0, 0.5, 1.5 and 3, and σ_relax − ΔI = σ_inst + ΔI in each case. The bracket widens fast with g: at g = 3, ΔI = 0.80, σ_relax = 4.0 and σ_inst = +2.4. The escape path then produces entropy, and the upper bound on ln r is positive, which is vacuous.
What could have failed. The circulation’s contribution could have differed between relaxation and escape, which would break the common σ_a. Orthogonality makes it identical on both.
What this does not say
- That the longest-lived attractor is the most dissipative. The bound ranks lifetimes by path entropies only where it is tight. Falasco & Esposito show that steady dissipation and lifetime agree only “under some peculiar conditions” (THEORY-1270).
- That the bounds are useful far from equilibrium. In the Schlögl model, where bistability exists only far from equilibrium, the path entropies “are very large, thus making the bound (160) very loose” (Falasco & Esposito §VII.A.1). Santolin et al.’s numerical case is a weak shear.
- That the path entropies are finite as stated. Between fixed points of a nonequilibrium dynamics the integrals diverge, because a maintained fixed point dissipates at a steady rate forever. The endpoints must be taken on balls of radius O(Ω^{−1/2}) (Falasco & Esposito, appendix A), so the bounds carry a cutoff.
- Anything about prefactors. Only the exponential order is controlled; at moderate Ω the prefactor can change the rate by orders of magnitude, and for the Schlögl model it even decides which state is favoured at small V (THEORY-1273).
- Limit cycles or chaotic attractors, which both papers set aside.