Long-time convergence of an adaptive biasing force method

We propose a proof of convergence of an adaptive method used in molecular dynamics to compute free energy profiles (see Darve and Porohille 2001 J. Chem. Phys. 115 9169–83, Hénin and Chipot 2004 J. Chem. Phys. 121 2904–14, Lelièvre et al J. Chem. Phys. 126 134111). Mathematically, it amounts to studying the long-time behaviour of a stochastic process which satisfies a nonlinear stochastic differential equation, where the drift depends on conditional expectations of some functionals of the process. We use entropy techniques to prove exponential convergence to the stationary state.

[1]  Maria G. Westdickenberg,et al.  A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit , 2009 .

[2]  T. Lelièvre.  A general two-scale criteria for logarithmic Sobolev inequalities , 2009 .

[3]  Raphael Roux,et al.  Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process , 2009 .

[4]  G. Ciccotti,et al.  Projection of diffusions on submanifolds: Application to mean force computation , 2008 .

[5]  Christophe Chipot,et al.  Free Energy Calculations , 2008 .

[6]  T. Lelièvre,et al.  Computation of free energy profiles with parallel adaptive dynamics. , 2006, The Journal of chemical physics.

[7]  T. Lelièvre,et al.  Computation of free energy differences through nonequilibrium stochastic dynamics: The reaction coordinate case , 2006, J. Comput. Phys..

[8]  Maria G. Reznikoff,et al.  A new criterion for the logarithmic Sobolev inequality and two applications , 2007 .

[9]  A. Laio,et al.  Equilibrium free energies from nonequilibrium metadynamics. , 2006, Physical review letters.

[10]  Christophe Chipot,et al.  Exploring the free-energy landscape of a short peptide using an average force. , 2005, The Journal of chemical physics.

[11]  C. Chipot,et al.  Overcoming free energy barriers using unconstrained molecular dynamics simulations. , 2004, The Journal of chemical physics.

[12]  G. Burton.  TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

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

[14]  C. Villani.  Topics in Optimal Transportation , 2003 .

[15]  Eric F Darve,et al.  Calculating free energies using average force , 2001 .

[16]  D. Landau,et al.  Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Giuseppe Toscani,et al.  ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS , 2001 .

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

[19]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[20]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[21]  Djalil CHAFAÏ,et al.  Sur les in'egalit'es de Sobolev logarithmiques , 2000 .

[22]  S. Bobkov,et al.  Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities , 1999 .

[23]  Michiel Sprik,et al.  Free energy from constrained molecular dynamics , 1998 .

[24]  W. Briels,et al.  THE CALCULATION OF FREE-ENERGY DIFFERENCES BY CONSTRAINED MOLECULAR-DYNAMICS SIMULATIONS , 1998 .

[25]  C. Jarzynski.  Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach , 1997, cond-mat/9707325.

[26]  M.G.B. Drew,et al.  The art of molecular dynamics simulation , 1996 .

[27]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[28]  D. Stroock,et al.  Logarithmic Sobolev inequalities and stochastic Ising models , 1987 .

[29]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[30]  M. Émery,et al.  Hypercontractivité de semi-groupes de diffusion , 1984 .

[31]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[32]  E. Davies.  Spectral properties of metastable Markov semigroups , 1983 .

[33]  E. Davies,et al.  Metastable States of Symmetric Markov Semigroups II , 1982 .

[34]  E. Davies,et al.  Metastable States of Symmetric Markov Semigroups I , 1982 .

[35]  E. Davies.  Dynamical stability of metastable states , 1982 .

[36]  P. Meyer,et al.  Sur les inegalites de Sobolev logarithmiques. I , 1982 .

[37]  L. Gross.  LOGARITHMIC SOBOLEV INEQUALITIES. , 1975 .