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Metastable transitions in inertial Langevin systems: What can be different from the overdamped case?

Part of: Statistical mechanics, structure of matter Stochastic analysis

Published online by Cambridge University Press:  19 September 2018

ANDRE N. SOUZA  and
MOLEI TAO
ANDRE N. SOUZA
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA emails: andre.souza@gatech.edu; mtao@gatech.edu
MOLEI TAO
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA emails: andre.souza@gatech.edu; mtao@gatech.edu
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Abstract

Metastable transitions in Langevin dynamics can exhibit rich behaviours that are markedly different from its overdamped limit. In addition to local alterations of the transition path geometry, more fundamental global changes may exist. For instance, when the dissipation is weak, heteroclinic connections that exist in the overdamped limit do not necessarily have a counterpart in the Langevin system, potentially leading to different transition rates. Furthermore, when the friction coefficient depends on the velocity, the overdamped limit no longer exists, but it is still possible to efficiently find instantons. The approach, we employed for these discoveries, was based on (i) a simple rewriting of the Freidlin–Wentzell action in terms of time-reversed dynamics and (ii) an adaptation of the string method, which was originally designed for gradient systems, to this specific non-gradient system.

Keywords

Freidlin–Wentzell large deviation theoryinstantoninertial Langevin equationunderdamped dynamicsmatrix-valued variable friction coefficient

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 82-08: Computational methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 5: Stochastic Analysis in Applications , October 2019 , pp. 830 - 852
Copyright
© Cambridge University Press 2018 

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Footnotes

†

M. T. is partially supported by NSF grant DMS-1521667 and ECCS-1829821 and A. S. is partially supported by DMS-1344199.

References

Bouchet, F. & Reygner, J. (2016) Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes Ann. Henri Poincaré 17(12), 3499–3532. doi: 10.1007/s00023-016-0507-4.CrossRefGoogle Scholar
Boyd, J. (2001) Chebyshev and Fourier Spectral Methods, 2nd ed., Dover, New York.Google Scholar
Dahiya, D. & Cameron, M. (2018) Ordered line integral methods for computing the quasi-potential. J. Sci. Comput. 75(3), 1351–1384. doi: 10.1007/s10915-017-0590-9.CrossRefGoogle Scholar
Dembo, A. & Zeitouni, O. (2010) Large Deviations Techniques and Applications, 2nd ed., Springer, New York.CrossRefGoogle Scholar
D’Orsogna, M. R., Chuang, Y.-L., Bertozzi, A. L. & Chayes, L.S. (2006) Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96(10), 104302.CrossRefGoogle Scholar
E, W., Ren, W. & Vanden-Eijnden, E. (2002) String method for the study of rare events. Phys. Rev. B 66, 052301. doi: 10.1103/PhysRevB.66.052301.CrossRefGoogle Scholar
E, W., Ren, W. & Vanden-Eijnden, E. (2004) Minimum action method for the study of rare events. Commun. Pure Appl. Math. 57(5), 637–656. doi: 10.1002/cpa.20005.CrossRefGoogle Scholar
E, W., Ren, W. & Vanden-Eijnden, E. (2007) Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys. 126(16), 164103. doi: 10.1063/1.2720838.CrossRefGoogle ScholarPubMed
E, W. & Zhou, X. (2011) The gentlest ascent dynamics. Nonlinearity 24(6), 1831. http://stacks.iop.org/0951-7715/24/i=6/a=008.CrossRefGoogle Scholar
Eggleton, P. P., Kiseleva, L. G. & Hut, P. (1998) The equilibrium tide model for tidal friction. Astrophys. J. 499(2), 853.CrossRefGoogle Scholar
Forgoston, E. & Moore, R. O. (accepted) A primer on noise-induced transitions in applied dynamical systems. SIAM Rev.Google Scholar
Freidlin, M. & Hu, W. (2011) Smoluchowski-Kramers approximation in the case of variable friction. J. Math. Sci. 179, 184.CrossRefGoogle Scholar
Freidlin, M. & Wentzell, A. (2012) Random Perturbations of Dynamical Systems, 3rd ed., Springer, New York.CrossRefGoogle Scholar
Freidlin, M., Hu, W. & Wentzell, A. (2013) Small mass asymptotic for the motion with vanishing friction. Stoch. Process. Appl. 123(1), 45–75. doi: 10.1016/j.spa.2012.08.013.CrossRefGoogle Scholar
Gardiner, C. W. (1985) Handbook of Stochastic Methods, 2nd ed., Springer, New York.Google Scholar
Grafke, T., Grauer, R. & Schäfer, T. (2015) The instanton method and its numerical implementation in fluid mechanics. J. Phys. A: Math. Theor. 48(33), 333001. http://stacks.iop.org/1751-8121/48/i=33/a=333001.CrossRefGoogle Scholar
Grafke, T., Schäfer, T. & Vanden-Eijnden, E. (2017) Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools Springer, New York, pp. 17–55. doi: 10.1007/978-1-4939-6969-2_2.Google Scholar
Greengard, L. (1991) Spectral integration and two-point boundary value problems. SIAM J. Numer. Anal. 28(4), 1071–1080. doi: 10.1137/0728057.CrossRefGoogle Scholar
Herzog, D. P., Hottovy, S. & Volpe, G. (2016) The small-mass limit for Langevin dynamics with unbounded coefficients and positive friction. J. Stat. Phys. 163, 659–673. doi: 10.1007/s10955-016-1498-8.CrossRefGoogle Scholar
Heymann, M. & Vanden-Eijnden, E. (2008) The geometric minimum action method: a least action principle on the space of curves. Commun. Pure Appl. Math. 61(8), 1052–1117. doi: 10.1002/cpa.20238.CrossRefGoogle Scholar
Heymann, M. & Vanden-Eijnden, E. (2008) Pathways of maximum likelihood for rare events in nonequilibrium systems: application to nucleation in the presence of shear. Phys. Rev. Lett. 100(14), 140601.CrossRefGoogle ScholarPubMed
Hottovy, S., McDaniel, A., Volpe, G. & Wehr, J. (2015) The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. Commun. Math. Phys. 336(3), 1259–1283.CrossRefGoogle Scholar
Levine, H., Rappel, W.-J. & Cohen, I. (2000) Self-organization in systems of self-propelled particles. Phys. Rev. E 63(1), 017101.CrossRefGoogle ScholarPubMed
Lindley, B. S. & Schwartz, I. B. (2013) An iterative action minimizing method for computing optimal paths in stochastic dynamical systems. Phys. D: Nonlinear Phenomena 255, 22–30.CrossRefGoogle Scholar
Maragliano, L., Fischer, A., Vanden-Eijnden, E. & Ciccotti, G. (2006) String method in collective variables: minimum free energy paths and isocommittor surfaces. J. Chem. Phys. 125(2), 024106. doi: 10.1063/1.2212942.CrossRefGoogle ScholarPubMed
Mardling, R. A. & Lin, D. (2002) Calculating the tidal, spin, and dynamical evolution of extrasolar planetary systems. Astrophys. J. 573(2), 829.CrossRefGoogle Scholar
Nelson, E. (2001) Dynamical Theories of Brownian Motion, 2nd ed., Princeton University Press, Princeton, NJ.Google Scholar
Newhall, K. A. & Vanden-Eijnden, E. (2013) 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. doi: 10.1063/1.4804070.CrossRefGoogle Scholar
Nualart, D. (2006) The Malliavin Calculus and Related Topics, Vol. 1995, Springer, New York.Google Scholar
Olsson, H., Åström, K. J., de Wit, C. C., Gäfvert, M. & Lischinsky, P. (1998) Friction models and friction compensation. Eur. J. Control 4(3), 176–195. doi: 10.1016/S0947-3580(98)70113-X.CrossRefGoogle Scholar
Pavliotis, G. A. (2014) Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations, 1st ed., Springer-Verlag, New York.CrossRefGoogle Scholar
Ren, W. & Vanden-Eijnden, E. (2013) A climbing string method for saddle point search. J. Chem. Phys. 138, 134105.CrossRefGoogle ScholarPubMed
Rogers, L. C. G. & Williams, D. (1994) Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus, Vol. 2, Cambridge University Press, Cambridge, UK.Google Scholar
Tao, M. (2018) Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions. Phys. D 363(15), 1–17.CrossRefGoogle Scholar
Tepper, H. L. & Voth, G. A. (2006) Mechanisms of passive ion permeation through lipid bilayers: insights from simulations. J. Phys. Chem. B 110(42), 21327–21337. doi: 10.1021/jp064192h.CrossRefGoogle ScholarPubMed
Titulaer, U. (1978) A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high-friction case. Phys. A: Stat. Mech. Appl. 91(3), 321–344. doi: 10.1016/0378-4371(78)90182-6.CrossRefGoogle Scholar
Urbakh, M., Klafter, J., Gourdon, D. & Israelachvili, J. (2004) The nonlinear nature of friction. Nature 430, 525. doi: 10.1038/nature02750.CrossRefGoogle ScholarPubMed
Vanden-Eijnden, E. & Heymann, M. (2008) The geometric minimum action method for computing minimum energy paths. J. Chem. Phys. 128, 061103.CrossRefGoogle ScholarPubMed
Villani, C. (2009) Hypocoercivity, Issue No. 949–951, American Mathematical Society. [online]CrossRefGoogle Scholar
Viswanath, D. (1991) Spectral integration of linear boundary value problems. J. Comput. Appl. Math. 290, 159–173. doi: 10.1016/j.cam.2015.04.043.CrossRefGoogle Scholar
Wan, X. (2011) An adaptive high-order minimum action method. J. Comput. Phys. 230(24), 8669–8682. doi: 10.1016/j.jcp.2011.08.006.CrossRefGoogle Scholar
Zhou, X., Ren, W. & E, W. (2008) Adaptive minimum action method for the study of rare events. J. Chem. Phys. 128(10), 104111. doi: 10.1063/1.2830717.CrossRefGoogle Scholar