[Taux de sortie d’Eyring-Kramers pour la dynamique de Langevin sur-amortie : le cas des points-selles sur la frontière]
Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics
| \[ dX_t=-\nabla f(X_t) \, dt +\sqrt{h} \, dB_t \] |
and let $\Omega \subset \mathbb{R}^d $ be the basin of attraction of a local minimum of $f: \mathbb{R}^d \rightarrow \mathbb{R}$. Up to a small perturbation of $\Omega $ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $\Omega $ through each of the saddle points of $f$ on $\partial \Omega $ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \rightarrow 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics.
On considère la dynamique de Langevin sur-amortie
| \[ dX_t = -\nabla f(X_t) \, dt + \sqrt{h} \, dB_t \] |
et $\Omega \subset \mathbb{R}^d$, le bassin d’attraction d’un minimum local de $f: \mathbb{R}^d \rightarrow \mathbb{R}$. Quitte à légèrement perturber $\Omega $ pour en lisser le bord, nous montrons que les taux de sortie du processus $(X_t)_{t \ge 0}$ de $\Omega $ par chacun des points-selles de $f$ sur $\partial \Omega $ peuvent être paramétrés par les lois d’Eyring-Kramers, dans la limite $h \rightarrow 0$. Ce résultat fournit une base mathématique solide aux modèles de sauts markoviens qui sont utilisés pour décrire l’évolution des systèmes moléculaires, ainsi qu’à certaines méthodes numériques qui s’appuient sur ces modèles pour échantillonner efficacement des trajectoires métastables de la dynamique de Langevin sur-amortie.
Accepté le :
Publié le :
Keywords: Overdamped Langevin, Eyring-Kramers law, the exit problem, semi-classical analysis
Mots-clés : Dynamique de Langevin sur-amortie, loi d’Eyring-Kramers, problème de sortie, analyse semi-classique
Tony Lelièvre 1 ; Dorian Le Peutrec 2 ; Boris Nectoux 3
CC-BY 4.0
@article{JEP_2025__12__881_0,
author = {Tony Leli\`evre and Dorian Le Peutrec and Boris Nectoux},
title = {Eyring-Kramers exit rates for the~overdamped {Langevin} dynamics: the~case~with saddle points on the boundary},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {881--982},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.303},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.303/}
}
TY - JOUR AU - Tony Lelièvre AU - Dorian Le Peutrec AU - Boris Nectoux TI - Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 881 EP - 982 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.303/ DO - 10.5802/jep.303 LA - en ID - JEP_2025__12__881_0 ER -
%0 Journal Article %A Tony Lelièvre %A Dorian Le Peutrec %A Boris Nectoux %T Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 881-982 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.303/ %R 10.5802/jep.303 %G en %F JEP_2025__12__881_0
Tony Lelièvre; Dorian Le Peutrec; Boris Nectoux. Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 881-982. doi : 10.5802/jep.303. https://jep.centre-mersenne.org/articles/10.5802/jep.303/
[1] - “Mathematical analysis of temperature accelerated dynamics”, Multiscale Model. Simul. 12 (2014) no. 1, p. 290-317 | DOI | MR | Zbl
[2] - “Efficient annealing of radiation damage near grain boundaries via interstitial emission”, Science 327 (2010) no. 5973, p. 1631-1634 | DOI
[3] - “Kramers’ law: validity, derivations and generalisations”, Markov Process. Related Fields 19 (2013) no. 3, p. 459-490 | MR | Zbl
[4] - “The Eyring-Kramers law for Markovian jump processes with symmetries”, J. Theoret. Probab. 29 (2016) no. 4, p. 1240-1279 | DOI | Zbl
[5] - “The Eyring-Kramers law for potentials with nonquadratic saddles”, Markov Process. Related Fields 16 (2010) no. 3, p. 549-598 | MR | Zbl
[6] - “On soft capacities, quasi-stationary distributions and the pathwise approach to metastability”, J. Statist. Phys. 181 (2020) no. 3, p. 1052-1086 | DOI | MR | Zbl
[7] - “Complete asymptotics for solution of singularly perturbed dynamical systems with single well potential”, Mathematics 8 (2020) no. 6, p. 949 | DOI
[8] - “Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times”, J. Eur. Math. Soc. (JEMS) 6 (2004) no. 4, p. 399-424 | DOI | MR | Zbl
[9] - “Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues”, J. Eur. Math. Soc. (JEMS) 7 (2005) no. 1, p. 69-99 | DOI | MR | Zbl
[10] - Metastability. A potential-theoretic approach, Grundlehren Math. Wissen., vol. 351, Springer, Cham, 2015 | DOI | MR | Zbl
[11] - Introduction to differential topology, Cambridge University Press, Cambridge, 1982 | MR | Zbl
[12] - “The mixed problem for Laplace’s equation in a class of Lipschitz domains”, Comm. Partial Differential Equations 19 (1994) no. 7-8, p. 1217-1233 | DOI | MR | Zbl
[13] - “Computing the asymptotic spectrum for networks representing energy landscapes using the minimum spanning tree”, Netw. Heterog. Media 9 (2014) no. 3, p. 383-416 | DOI | MR | Zbl
[14] - “Quasi-stationary distributions and diffusion models in population dynamics”, Ann. Probab. 37 (2009) no. 5, p. 1926-1969 | DOI | MR | Zbl
[15] - “Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions”, in Séminaire de Probabilités XLIX, Lect. Notes in Math., vol. 2215, Springer, Cham, 2018, p. 165-182 | DOI | Zbl
[16] - “A cohomology complex for manifolds with boundary”, Topol. Methods Nonlinear Anal. 5 (1995) no. 2, p. 325-340 | DOI | MR | Zbl
[17] - Introduction to the theory of linear partial differential equations, Studies in Math. and its Appl., vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982 | Zbl
[18] - Quasi-stationary distributions. Markov chains, diffusions and dynamical systems, Probability and its Appl., Springer, Heidelberg, 2013 | DOI | MR | Zbl
[19] - Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987 | MR | Zbl
[20] - “Exponential leveling for stochastically perturbed dynamical systems”, SIAM J. Math. Anal. 13 (1982) no. 4, p. 532-540 | DOI | MR | Zbl
[21] - “On the exponential exit law in the small parameter exit problem”, Stochastics 8 (1983) no. 4, p. 297-323 | DOI | MR | Zbl
[22] - “On the asymptotic relation between equilibrium density and exit measure in the exit problem”, Stochastics 12 (1984) no. 3-4, p. 303-330 | DOI | MR | Zbl
[23] - “Recent progress on the small parameter exit problem”, Stochastics 20 (1987) no. 2, p. 121-150 | DOI | MR | Zbl
[24] - “Large deviations results for the exit problem with characteristic boundary”, J. Math. Anal. Appl. 147 (1990) no. 1, p. 134-153 | DOI | MR | Zbl
[25] - “Small noise spectral gap asymptotics for a large system of nonlinear diffusions”, J. Spectral Theory 7 (2017) no. 4, p. 939-984 | DOI | MR | Zbl
[26] - “Jump Markov models and transition state theory: the quasi-stationary distribution approach”, Faraday Discussions 195 (2017), p. 469-495 | DOI
[27] - “Sharp asymptotics of the first exit point density”, Ann. PDE 5 (2019) no. 1, article ID 5, 174 pages | DOI | MR | Zbl
[28] - “The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 1”, J. Math. Pures Appl. (9) 138 (2020), p. 242-306 | DOI | MR | Zbl
[29] - Spectral asymptotics in the semi-classical limit, London Math. Soc. Lect. Note Series, vol. 268, Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl
[30] - “The exponential leveling and the Ventcel-Freidlin ‘minimal action’ function”, J. Analyse Math. 54 (1990) no. 1, p. 99-111 | DOI | MR | Zbl
[31] - Partial differential equations, Graduate Studies in Math., vol. 19, American Mathematical Society, Providence, RI, 2010 | DOI | Zbl
[32] - Measure theory and fine properties of functions, Textbooks in Math., CRC Press, Boca Raton, FL, 2015 | DOI | MR | Zbl
[33] - “Autonomous basin climbing method with sampling of multiple transition pathways: application to anisotropic diffusion of point defects in hcp Zr”, J. Phys.: Condensed Matter 26 (2014), p. 365402
[34] - Random perturbations of dynamical systems, Grundlehren Math. Wissen., vol. 260, Springer, Heidelberg, 2012 | DOI | MR | Zbl
[35] - Elliptic partial differential equations of second order, Classics in Math., Springer-Verlag, Berlin, 2001 | DOI | Zbl
[36] - “Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds”, J. Math. Sci. 172 (2011) no. 3, p. 347-400 | DOI | Zbl
[37] - Differential topology, vol. 370, American Mathematical Society, Providence, RI, 2010 | DOI | MR | Zbl
[38] - “Reaction-rate theory: fifty years after Kramers”, Rev. Modern Phys. 62 (1990) no. 2, p. 251-342 | DOI | MR
[39] - Semi-classical analysis for the Schrödinger operator and applications, Lect. Notes in Math., vol. 1336, Springer-Verlag, Berlin, 1988 | DOI | MR | Zbl
[40] - Spectral theory and its applications, Cambridge Studies in Advanced Math., vol. 139, Cambridge University Press, Cambridge, 2013 | DOI | MR | Zbl
[41] - “Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach”, Mat. Contemp. 26 (2004), p. 41-85 | MR | Zbl
[42] - Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary, Mém. Soc. Math. France (N.S.), vol. 105, Société Mathématique de France, Paris, 2006 | Numdam | Zbl
[43] - “Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation”, Ann. Inst. H. Poincaré Phys. Théor. 42 (1985) no. 2, p. 127-212 | Numdam | MR | Zbl
[44] - “Puits multiples en mécanique semi-classique, IV: Etude du complexe de Witten”, Comm. Partial Differential Equations 10 (1985) no. 3, p. 245-340 | DOI | MR | Zbl
[45] - “Tunnel effect and symmetries for Kramers-Fokker-Planck type operators”, J. Inst. Math. Jussieu 10 (2011) no. 3, p. 567-634 | DOI | MR | Zbl
[46] - “Asymptotics of the spectral gap with applications to the theory of simulated annealing”, J. Funct. Anal. 83 (1989) no. 2, p. 333-347 | DOI | MR | Zbl
[47] - “Metastability for parabolic equations with drift: Part I”, Indiana Univ. Math. J. 64 (2015), p. 875-913 | DOI | MR | Zbl
[48] - “Metastability for parabolic equations with drift: part II. The quasilinear case”, Indiana Univ. Math. J. 66 (2017), p. 315-360 | DOI | MR | Zbl
[49] - “On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains”, Indiana Univ. Math. J. 58 (2009) no. 5, p. 2043-2071 | DOI | MR | Zbl
[50] - Riemannian geometry and geometric analysis, Universitext, Springer-Verlag, Berlin, 2008 | DOI | MR | Zbl
[51] - “Elliptic perturbation of a first order operator with a singular point of attracting type”, Indiana Univ. Math. J. 27 (1978) no. 6, p. 935-952 | DOI | MR | Zbl
[52] - “On elliptic singular perturbation problems with turning points”, SIAM J. Math. Anal. 10 (1979) no. 3, p. 447-455 | DOI | MR | Zbl
[53] - “Dirichlet’s and Thomson’s Principles for Non-selfadjoint Elliptic Operators with Application to Non-reversible Metastable Diffusion Processes”, Arch. Rational Mech. Anal. 231 (2019) no. 2, p. 887-938 | DOI | MR | Zbl
[54] - “A Morse complex on manifolds with boundary”, Geom. Dedicata 153 (2011) no. 1, p. 47-57 | DOI | MR | Zbl
[55] - “A mathematical formalization of the parallel replica dynamics”, Monte Carlo Methods Appl. 18 (2012) no. 2, p. 119-146 | DOI | MR | Zbl
[56] - “Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian”, Ann. Fac. Sci. Toulouse Math. (6) 19 (2010) no. 3-4, p. 735-809 | DOI | Numdam | MR | Zbl
[57] - “Sharp spectral asymptotics for non-reversible metastable diffusion processes”, Probab. Math. Phys. 1 (2020) no. 1, p. 3-53 | DOI | MR | Zbl
[58] - “Small eigenvalues of the Witten Laplacian with Dirichlet boundary conditions: the case with critical points on the boundary”, Anal. PDE 14 (2021) no. 8, p. 2595-2651 | DOI | MR | Zbl
[59] - “Precise Arrhenius law for -forms: the Witten Laplacian and Morse-Barannikov complex”, Ann. Henri Poincaré 14 (2013) no. 3, p. 567-610 | DOI | MR | Zbl
[60] - Bar codes of persistent cohomology and Arrhenius law for -forms, Astérisque, vol. 450, Société Mathématique de France, Paris, 2024 | DOI | Zbl
[61] - “Non-reversible metastable diffusions with Gibbs invariant measure I: Eyring–Kramers formula”, Probab. Theory Related Fields 182 (2022) no. 3, p. 849-903 | DOI | MR | Zbl
[62] - “Mathematical foundations of accelerated molecular dynamics methods”, in Handbook of materials modeling (W. Andreoni & S. Yip, eds.), 2018 | DOI
[63] - “Exit event from a metastable state and Eyring-Kramers law for the overdamped Langevin dynamics”, in Stochastic dynamics out of equilibrium, Springer Proc. Math. Stat., vol. 282, Springer, Cham, 2019, p. 331-363 | DOI | Zbl
[64] - “The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2”, Stochastic Partial Differ. Equ. Anal. Comput. 10 (2022), p. 317-357 | MR | Zbl
[65] - “Low temperature asymptotics for quasistationary distributions in a bounded domain”, Anal. PDE 8 (2015) no. 3, p. 561-628 | DOI | MR | Zbl
[66] - “Spectra, exit times and long time asymptotics in the zero-white-noise limit”, Stochastics 55 (1995) no. 1-2, p. 1-20 | Zbl
[67] - “The exit problem for randomly perturbed dynamical systems”, SIAM J. Appl. Math. 33 (1977) no. 2, p. 365-382 | DOI | MR | Zbl
[68] - “The exit problem: a new approach to diffusion across potential barriers”, SIAM J. Appl. Math. 36 (1979) no. 3, p. 604-623 | MR | Zbl
[69] - “Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape”, Ann. Probab. 42 (2014) no. 5, p. 1809-1884 | DOI | MR | Zbl
[70] - “About small eigenvalues of Witten Laplacian”, Pure Appl. Anal. 1 (2019) no. 2, p. 149-206 | DOI | MR | Zbl
[71] - “Comportement de spectres d’opérateurs de Schrödinger à basse température”, Bull. Sci. Math. 119 (1995) no. 6, p. 529-554 | MR | Zbl
[72] - Topology from the differentiable viewpoint, Princeton Landmarks in Math., Princeton University Press, Princeton, NJ, 1997 | MR | Zbl
[73] - “Closing the gap between experiment and theory: crystal growth by temperature accelerated dynamics”, Phys. Rev. Lett. 87 (2001), article ID 126101, 4 pages | DOI
[74] - “A direct approach to the exit problem”, SIAM J. Appl. Math. 50 (1990) no. 2, p. 595-627 | DOI | MR | Zbl
[75] - “Mean exit time for the overdamped Langevin process: the case with critical points on the boundary”, Comm. Partial Differential Equations 46 (2021) no. 9, p. 1789-1829 | DOI | Zbl
[76] - “Everything you wanted to know about Markov state models but were afraid to ask”, Methods 52 (2010) no. 1, p. 99-105 | DOI
[77] - “Recent advances in accelerated molecular dynamics methods: Theory and applications”, in Comprehensive Computational Chemistry (M. Yáñez & R. J. Boyd, eds.), vol. 3, Elsevier, 2024, p. 360-383 | DOI
[78] - “Perturbed dynamical systems with an attracting singularity and weak viscosity limits in Hamilton-Jacobi equations”, Trans. Amer. Math. Soc. 317 (1990) no. 2, p. 723-748 | DOI | MR | Zbl
[79] - Conformational dynamics: modelling, theory, algorithm and application to biomolecules, Habilitation dissertation, Free University Berlin, 1998
[80] - Metastability and Markov state models in molecular dynamics, Courant Lecture Notes in Math., vol. 24, American Mathematical Society, Providence, RI, 2013 | DOI | MR | Zbl
[81] - Hodge decomposition—a method for solving boundary value problems, Lect. Notes in Math., vol. 1607, Springer-Verlag, Berlin, 1995 | DOI | MR | Zbl
[82] - “Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: Asymptotic expansions”, Ann. Inst. H. Poincaré Phys. Théor. 38 (1983) no. 3, p. 295-308 | Numdam | Zbl
[83] - “Temperature-accelerated dynamics for simulation of infrequent events”, J. Chem. Phys. 112 (2000) no. 21, p. 9599-9606 | DOI
[84] - “Simulation of growth of Cu on Ag(001) at experimental deposition rates”, Phys. Rev. B 66 (2002), article ID 205415, 10 pages | DOI
[85] - “Exponential asymptotics in the small parameter exit problem”, Nagoya Math. J. 144 (1996), p. 137-154 | DOI | MR | Zbl
[86] - “Asymptotic behaviors on the small parameter exit problems and the singularly perturbation problems”, Ryukyu Math. J. 14 (2001), p. 79-118 | MR | Zbl
[87] - “Quelques propriétés globales des variétés différentiables”, Comment. Math. Helv. 28 (1954) no. 1, p. 17-86 | DOI | MR | Zbl
[88] - “From metadynamics to dynamics”, Phys. Rev. Lett. 111 (2013), article ID 230602, 5 pages | DOI
[89] - “Some characterizations of normal and perfectly normal spaces”, Duke Math. J. 19 (1952) no. 2, p. 289-292 | DOI | MR | Zbl
[90] - “A method for accelerating the molecular dynamics simulation of infrequent events”, J. Chem. Phys. 106 (1997) no. 11, p. 4665-4677 | DOI
[91] - “Introduction to the kinetic Monte Carlo method”, Radiation Effects in Solids, Springer, NATO Publishing Unit (2005)
[92] - Energy landscapes: applications to clusters, biomolecules and glasses, Cambridge Molecular Science, Cambridge University Press, Cambridge, 2004 | DOI
[93] - “Supersymmetry and Morse theory”, J. Differential Geom. 17 (1982) no. 4, p. 661-692 | MR | Zbl
Cité par Sources :