Convergence of metadynamics: Discussion of the adiabatic hypothesis

By drawing a parallel between metadynamics and self interacting models for polymers, we study the longtime convergence of the original metadynamics algorithm in the adiabatic setting, namely when the dynamics along the collective variables decouples from the dynamics along the other degrees of freedom. We also discuss the bias which is introduced when the adiabatic assumption does not holds.

[1]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[2]  Alessandro Laio,et al.  Metadynamics convergence law in a multidimensional system. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  I. Ibragimov,et al.  Independent and stationary sequences of random variables , 1971 .

[4]  S. Meyn,et al.  Stability of Markovian processes II: continuous-time processes and sampled chains , 1993, Advances in Applied Probability.

[5]  Alessandro Laio,et al.  Multidimensional view of amyloid fibril nucleation in atomistic detail. , 2012, Journal of the American Chemical Society.

[6]  S. Meyn,et al.  Exponential and Uniform Ergodicity of Markov Processes , 1995 .

[7]  A. Laio,et al.  Escaping free-energy minima , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[8]  A. Laio,et al.  Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science , 2008 .

[9]  Gersende Fort,et al.  Self-healing umbrella sampling: convergence and efficiency , 2017, Stat. Comput..

[10]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

[11]  Simone Marsili,et al.  Self-healing umbrella sampling: a non-equilibrium approach for quantitative free energy calculations. , 2006, The journal of physical chemistry. B.

[12]  Convergence and Efficiency of Adaptive Importance Sampling Techniques with Partial Biasing , 2016, 1610.09194.

[13]  T. Kurtz,et al.  Stochastic equations in infinite dimensions , 2006 .

[14]  Tobias Hurth,et al.  Invariant densities for dynamical systems with random switching , 2012, 1203.5744.

[15]  D. Landau,et al.  Efficient, multiple-range random walk algorithm to calculate the density of states. , 2000, Physical review letters.

[16]  C. Mouhot,et al.  HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS , 2010, 1005.1495.

[17]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[18]  Alessandro Laio,et al.  Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics. , 2003, Physical review letters.

[19]  Diffusivity bounds for 1D Brownian polymers , 2009, 0911.2356.

[20]  B. Tóth,et al.  Continuous time `true' self-avoiding random walk on Z , 2009, 0909.3863.

[21]  M. Benaim,et al.  Quantitative ergodicity for some switched dynamical systems , 2012, 1204.1922.

[22]  A. Laio,et al.  A novel approach to the investigation of passive molecular permeation through lipid bilayers from atomistic simulations. , 2012, The journal of physical chemistry. B.

[23]  A. Laio,et al.  Equilibrium free energies from nonequilibrium metadynamics. , 2006, Physical Review Letters.

[24]  Alexandre Genadot,et al.  Piecewise deterministic Markov process - recent results , 2013, 1309.6061.

[25]  F. Malrieu.  Some simple but challenging Markov processes , 2014, 1412.7516.

[26]  Self-repelling diffusions on a Riemannian manifold , 2015, 1505.05664.

[27]  Michele Parrinello,et al.  Well-tempered metadynamics converges asymptotically. , 2014, Physical review letters.

[28]  M. Parrinello,et al.  Well-tempered metadynamics: a smoothly converging and tunable free-energy method. , 2008, Physical review letters.

[29]  M. Benaim,et al.  Qualitative properties of certain piecewise deterministic Markov processes , 2012, 1204.4143.

[30]  M. Benaïm,et al.  Self-repelling diffusions via an infinite dimensional approach , 2014, 1408.6397.

[31]  A survey of random processes with reinforcement , 2007, math/0610076.

[32]  S. Puntanen.  Inequalities: Theory of Majorization and Its Applications, Second Edition by Albert W. Marshall, Ingram Olkin, Barry C. Arnold , 2011 .

[33]  T. Lelièvre,et al.  Free Energy Computations: A Mathematical Perspective , 2010 .

[34]  Grigorios A. Pavliotis,et al.  Multiscale Methods: Averaging and Homogenization , 2008 .