Journal of Nonlinear Science

, Volume 27, Issue 3, pp 1007–1042 | Cite as

Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory



This paper is concerned with the longtime dynamics of the nonlinear wave equation in one-space dimension,
$$\begin{aligned} u_{tt} - \kappa ^2 u_{xx} +V'(u) =0 \quad x\in [0,1] \end{aligned}$$
where \(\kappa >0\) is a parameter and V(u) is a potential bounded from below and growing at least like \(u^2\) as \(|u|\rightarrow \infty \). Infinite energy solutions of this equation preserve a natural Gibbsian invariant measure, and when the potential is double-welled, for example when \(V(u) = \tfrac{1}{4}(1-u^2)^2\), there is a regime such that two small disjoint sets in the system’s phase-space concentrate most of the mass of this measure. This suggests that the solutions to the nonlinear wave equation can be metastable over these sets, in the sense that they spend long periods of time in these sets and only rarely transition between them. Here, we quantify this phenomenon by calculating exactly via transition state theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. We also investigate numerically how the mean TST frequency compares to the rate at which a typical solution crosses this dividing surface. These numerical results suggest that the dynamics of the nonlinear wave equation is ergodic and rapidly mixing with respect to the Gibbs invariant measure when the parameter \(\kappa \) in small enough. In this case, successive transitions between the two regions are roughly uncorrelated and their dynamics can be coarse-grained to jumps in a two-state Markov chain whose rate can be deduced from the mean TST frequency. This is a regime in which the dynamics of the nonlinear wave equation displays a metastable behavior that is not fundamentally different from that observed in its stochastic counterpart in which random noise and damping terms are added to the equation. For larger \(\kappa \), however, the dynamics either stops being ergodic, or its mixing time becomes larger than the inverse of the TST frequency, indicating that successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.


Transition state theory Metastability Stochastic partial differential equation Effective dynamics Nonlinear wave equation 

Mathematics Subject Classification

37A60 60H15 82B05 



We would like to thank Weinan E and Gregor Kovačič for useful discussions. This work was supported in part by the NSF Grant DMS-1522767 (E. V.-E.).


  1. Ariel, G., Vanden-Eijnden, E.: Testing transition state theory on Kac–Zwanzig model. J. Stat. Phys. 126(1), 43–73 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. Barret, F.: Sharp asymptotics of metastable transition times for one dimensional SPDEs. Ann. Insit. Henri Poincaré Probab. Stat. 51(1), 129–166 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. Berglund, N., Gentz, B.: Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond. Electron. J. Probab. 18, 1–58 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166(1), 1–26 (1994)CrossRefMATHGoogle Scholar
  5. Bovier, A.: Metastability. Springer, Berlin (2015)CrossRefMATHGoogle Scholar
  6. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Relat. Fields 119(1), 99–161 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low lying spectra in reversible Markov chains. Commun. Math. Phys. 228(2), 219–255 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. 6, 399–424 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. Deng, Y.: Invariance of the Gibbs measure for the Benjamin–Ono equation. J. Eur. Math. Soc. 17(5), 1107–1198 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. Eyring, H.: The activated complex in chemical reactions. J. Chem. Phys. 3, 107–115 (1935)CrossRefGoogle Scholar
  11. Faris, W.G., Jona-Lasinio, G.: Large fluctuations for a nonlinear heat equation with noise. J. Phys. A Math. Gen. 15(10), 3025–3055 (1982)MathSciNetCrossRefMATHGoogle Scholar
  12. Flaschka, H.: Some geometry in high-dimensional spaces. Lecture Notes in Mathematics 527A (2010)Google Scholar
  13. Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Hamiltonian Systems, Volume 109 of Memoirs of the American Mathematical Society. American Mathematical Society, Providence (1994)Google Scholar
  14. Friedlander, L.: An invariant measure for the equation \(u_{tt}-u_{xx}+u^3=0\). Commun. Math. Phys. 98, 1–16 (1985)MathSciNetCrossRefGoogle Scholar
  15. Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. Horiuti, J.: On the statistical mechanical treatment of the absolute rate of chemical reaction. Bull. Chem. Soc. Jpn. 13(1), 210–216 (1938)CrossRefGoogle Scholar
  17. Huisinga, W., Meyn, S., Schütte, C.: Phase transitions and metastability in Markovian and molecular systems. Ann. Appl. Probab. 14(1), 419–458 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. Lebowitz, J.L., Rose, H.A., Speer, E.R.: Statistical-mechanics of the nonlinear Schrodinger-equation. J. Stat. Phys. 50(3–4), 657–687 (1988)MathSciNetCrossRefMATHGoogle Scholar
  19. McKean, H.P.: Geometry of differential space. Ann. Probab. 1(2), 197–206 (1973)MathSciNetCrossRefMATHGoogle Scholar
  20. McKean, H.P., Vaninsky, K.L.: Statistical mechanics of nonlinear wave equations. J. Math. Sci. 94(4), 1630–1634 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. Newhall, K.A., Vanden-Eijnden, E.: Averaged equation for energy diffusion on a graph reveals bifurcation diagram and thermally assisted reversal times in spin-torque driven nanomagnets. J. Appl. Phys. 113(18), 184105 (2013)CrossRefGoogle Scholar
  22. Oh, T.: Invariance of the Gibbs measure for the Schrödinger–Benjamin–Ono system. SIAM J. Math. Anal. 41(6), 2207–2225 (2010)CrossRefMATHGoogle Scholar
  23. Oh, T., Richards, G., Thomann, L.: On invariant Gibbs measures for the generalized KdV equations. Dyn. Partial Differ. Equ. 13(2), 133–153 (2016)MathSciNetCrossRefMATHGoogle Scholar
  24. Poincaré, H.: Calcul des Probabilités. Gauthier-Villars, Paris (1912)MATHGoogle Scholar
  25. Ryser, M.D., Nigam, N., Tupper, P.F.: On the well-posedness of the stochastic Allen–Cahn equation in two dimensions. J. Comput. Phys. 231, 2537–2550 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. Sinai, Y.G.: Introduction to Ergodic Theory. Princeton University Press, Princeton (1976)MATHGoogle Scholar
  27. Tal, F.A., Vanden-Eijnden, E.: Transition state theory and dynamical corrections in ergodic systems. Nonlinearlity 19, 501–509 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. Tzvetkov, N.: Invariant measures for the nonlinear Schrödinger equation on the disc. Dyn. Partial Differ. Equ. 3(2), 111–160 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. Tzvetkov, N.: Mesures invariantes pour l’équation de Schrödinger non linéaire. Ann. Inst. Fourier 58(7), 2543–2604 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. Vanden-Eijnden, E., Tal, F.A.: Transition state theory: variational formulation, dynamical corrections, and error estimates. J. Chem. Phys. 128, 184103 (2005)CrossRefGoogle Scholar
  31. Vanden-Eijnden, E., Westdickenberg, M.G.: Rare events in stochastic partial differential equations on large spatial domains. J. Stat. Phys. 131, 1023–1038 (2008)MathSciNetCrossRefMATHGoogle Scholar
  32. Wigner, E.: The transition state method. Trans. Farady Soc. 34, 29–41 (1938)CrossRefGoogle Scholar
  33. Zhidkov, P.E.: An invariant measure for a nonlinear wave equation. Nonlinear Anal. Theory Methods Appl. 22(3), 319–325 (1994)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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