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LIT-1981: The geometric minimum action method: A least action principle on the space of curves

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StatusActive — read in part 2026-11-03 (NOTE-1504, Skimmed: §§1, 2.1–2.2, 4.1 and 5), from the published text on a Peking University course page; worth reading as the standard computation of most likely transition paths in small-noise systems, which confirms QUESTION-071's note of 2026-11-02 for gradient drift with identity noise and shows, in the Maier–Stein model, that the noise path stays the reversed relaxation path until the drift is far from gradient.
Version2
Tagsmathematics, complex-systems, anthology-candidate
Cited as “paper” byNOTE-1504 — Heymann & Vanden-Eijnden — The geometric minimum action method
Cited as “source” byTHEORY-1309 — For a small-noise diffusion the Freidlin–Wentzell quasi-potential equals 2 inf ∫ |b| sin²(θ/2) ds over curves, θ the angle between curve and drift, and the minimizing curve is the path a noise-driven transition follows with probability tending to one; with gradient drift and identity noise that curve runs against the drift, the time-reversed relaxation path, while a far-from-gradient drift, as in the Maier–Stein model above β = 4, makes it depart from the heteroclinic orbit

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Heymann, M., Vanden-Eijnden, E. (2008). The geometric minimum action method: A least action principle on the space of curves. Communications on Pure and Applied Mathematics, 61(8), 1052-1117. doi:10.1002/cpa.20238.

Read in part on 2026-11-03 (NOTE-1504, Skimmed; THEORY-1309).

Key takeaways

  • V(x₁, x₂) = 2 inf ∫ |b|_a sin²(θ/2) ds over curves (eq. 1.18).
  • Gradient drift, identity noise: the escape path is the reversed relaxation path (p. 1058).
  • Maier–Stein: gradient only at β = 1, yet the path is the heteroclinic orbit up to β = 4 (§4.1).

Standing in the record

Computes the most likely transition paths of a noisy system. Bears on the note of 2026-11-02 on QUESTION-071, that a transition’s path, not its end state, can tell a noise-driven escape from a directed one.

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