Source
Falasco & Esposito (2025), LIT-1892, §IV.D (eqs. 132–138), §IV.E (eqs. 139–146), §VII.A.1 (eqs. 243–244, Fig. 4), read in NOTE-1472. Vellela & Qian (2009), LIT-1890, appendix A (eqs. A1–A11) and footnote 5, read in NOTE-1474.
What was actually shown
The barrier. For a one-step process in one dimension the exact quasi-potential satisfies dI_ss/dc = ln(Σr₋/Σr₊), while truncating the Kramers–Moyal expansion at second order gives dI_NLE/dc = −F/D = 2(Σr₋ − Σr₊)/(Σr₋ + Σr₊) (Falasco & Esposito eqs. 137–138; Vellela & Qian eqs. A7–A8). The two agree to first order where F = 0 and nowhere else. In the Schlögl model at k₋₂ = 3.5, the diffusion barrier falls short of the exact one by δ = 0.0001 at Δμ = 4 (near the bifurcation), 0.019 at 4.5, 0.19 at 5 and 0.50 at 5.3 (my computation; Fig. 4’s trend). At V ≈ 30, δ ≈ 0.2 is a factor e⁶ ≈ 400 in the escape rate (§VII.A.1).
The wrong winner. Which of two states dominates as V → ∞ is the sign of ∫ ln q between them for the master equation and of 2∫(q − 1)/(q + 1) for the Fokker–Planck approximation, q = w/v (Vellela & Qian eq. A11). At k = (3, 0.6, 0.25, 2.95), a = 1, b = 1.305 (footnote 5) I find +0.00166 and −0.00020: the master equation concentrates on the low state and the diffusion approximation on the high one.
The entropy production. The path-probability entropy of a nonlinear Langevin equation has mean F·D⁻¹·F (Falasco & Esposito eq. 144), which is zero at any fixed point (eq. 145). A maintained nonequilibrium state has a positive steady dissipation, ⟨σ̇_γ⟩ in eq. (225); the Langevin description misses it entirely, keeping only the relaxation part.
What could have failed. Diffusion and jump descriptions share the drift, the extrema and the curvatures at the extrema (Vellela & Qian eqs. A9–A10). The barrier between them could have agreed too; it does not, because the large-deviation form p ≍ e^{−VI} is not smooth on the jump scale 1/V (Falasco & Esposito eqs. 132–133).
What this does not say
- That diffusion approximations are useless. Near a fixed point, for Gaussian fluctuations (the linear noise approximation), and near a bifurcation where the barrier is small, they agree with the master equation to leading order.
- That every continuum description fails. In a continuous-space limit where jumps become infinitesimal the Einstein relation is restored and the Langevin description is accurate (Falasco & Esposito §V.A, not read).
- Anything beyond one dimension for the closed forms. The general-dimension statement rests on the review’s argument, not on a computation here.