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Partial differential equations and stochastic methods in molecular dynamics*

Published online by Cambridge University Press:  23 May 2016

Tony Lelièvre  and
Gabriel Stoltz
Tony Lelièvre
Affiliation:
Université Paris-Est, CERMICS (ENPC), INRIA, F-77455 Marne-la-Vallée, France E-mail: tony.lelievre@enpc.fr, gabriel.stoltz@enpc.fr
Gabriel Stoltz
Affiliation:
Université Paris-Est, CERMICS (ENPC), INRIA, F-77455 Marne-la-Vallée, France E-mail: tony.lelievre@enpc.fr, gabriel.stoltz@enpc.fr
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Abstract

The objective of molecular dynamics computations is to infer macroscopic properties of matter from atomistic models via averages with respect to probability measures dictated by the principles of statistical physics. Obtaining accurate results requires efficient sampling of atomistic configurations, which are typically generated using very long trajectories of stochastic differential equations in high dimensions, such as Langevin dynamics and its overdamped limit. Depending on the quantities of interest at the macroscopic level, one may also be interested in dynamical properties computed from averages over paths of these dynamics.

This review describes how techniques from the analysis of partial differential equations can be used to devise good algorithms and to quantify their efficiency and accuracy. In particular, a crucial role is played by the study of the long-time behaviour of the solution to the Fokker–Planck equation associated with the stochastic dynamics.

Type
Research Article
Information
Acta Numerica , Volume 25 , 01 May 2016 , pp. 681 - 880
Copyright
© Cambridge University Press, 2016 

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References

REFERENCES22

Abdulle, A., Vilmart, G. and Zygalakis, K. C. (2015), ‘Long time accuracy of Lie–Trotter splitting methods for Langevin dynamics’, SIAM J. Numer. Anal. 53, 1–16.Google Scholar
Akhmatskaya, E. and Reich, S. (2005), The targeted shadowing hybrid Monte Carlo (TSHMC) method. New Algorithms for Macromolecular Simulation, Springer, pp. 141–153.Google Scholar
Allen, R., Valeriani, C. and ten Wolde, P. (2009), ‘Forward flux sampling for rare event simulations’, J. Phys.-Condens. Mat. 21, 463102–463122.Google Scholar
Allen, R., Warren, P. and ten Wolde, P. (2005), ‘Sampling rare switching events in biochemical networks’, Phys. Rev. Lett. 94, 018104.CrossRefGoogle ScholarPubMed
Alrachid, H. and Lelièvre, T. (2015), ‘Long-time convergence of an adaptive biasing force method: Variance reduction by Helmholtz projection’, SMAI J. Comput. Math. 1, 55–82.CrossRefGoogle Scholar
Ambrosio, L., Fusco, N. and Pallara, D. (2000), Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publications.CrossRefGoogle Scholar
Andradóttir, S., Heyman, D. P. and Ott, T. J. (1993), ‘Variance reduction through smoothing and control variates for Markov chain simulations’, ACM Trans. Model. Comput. Simul. 3, 167–189.CrossRefGoogle Scholar
Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000), Sur les Inégalités de Sobolev Logarithmiques (in French), Société Mathématique de France.Google Scholar
Aristoff, D. and Lelièvre, T. (2014), ‘Mathematical analysis of temperature accelerated dynamics’, SIAM Multiscale Model. Simul. 12, 290–317.Google Scholar
Aristoff, D., Lelièvre, T. and Simpson, G. (2014), ‘The parallel replica method for simulating long trajectories of Markov chains’, App. Math. Res. Express 2, 332–352.Google Scholar
Arnold, A., Markowich, P., Toscani, G. and Unterreiter, A. (2001), ‘On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations’, Comm. Partial Diff. Equations 26, 43–100.CrossRefGoogle Scholar
Assaraf, R. and Caffarel, M. (1999), ‘Zero-variance principle for Monte Carlo algorithms’, Phys. Rev. Lett. 83, 4682–4685.Google Scholar
Assaraf, R., Jourdain, B., Lelièvre, T. and Roux, R. (2015), Computation of sensitivities for the invariant measure of a parameter dependent diffusion. arXiv:1509.01348Google Scholar
Bakry, D. and Emery, M. (1985), Diffusions hypercontractives. In Séminaire de Probabilités XIX, Vol. 1123 of Lecture Notes in Mathematics, Springer, pp. 177–206.Google Scholar
Bakry, D., Barthe, F., Cattiaux, P. and Guillin, A. (2008), ‘A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case’, Electron. Comm. Probab. 13, 60–66.CrossRefGoogle Scholar
Bakry, D., Gentil, I. and Ledoux, M. (2014), Analysis and Geometry of Markov Diffusion Operators, Springer.CrossRefGoogle Scholar
Balian, R. (2007), From Microphysics to Macrophysics: Methods and Applications of Statistical Physics, Vols I–II, Springer.Google Scholar
Banisch, R. and Hartmann, C. (2016), ‘A sparse Markov chain approximation of LQ-type stochastic control problems’, Math. Control Rel. Fields, to appear.Google Scholar
Barkema, G. and Mousseau, N. (1996), ‘Event-based relaxation of continuous disordered systems’, Phys. Rev. Lett. 77, 4358–4361.Google Scholar
Baskes, M. I. (1992), ‘Modified embedded-atom potentials for cubic materials and impurities’, Phys. Rev. B 46, 2727–2742.CrossRefGoogle ScholarPubMed
Benaïm, M. and Bréhier, C.-E. (2016), Convergence of adaptive biasing potential methods for diffusions. arXiv:1603.08088Google Scholar
Berglund, N. (2013), ‘Kramers’ law: Validity, derivations and generalisations’, Markov Process. Rel. Fields 19, 459–490.Google Scholar
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2015), ‘Macroscopic fluctuation theory’, Rev. Mod. Phys. 87, 593–636.Google Scholar
Bhattacharya, R. N. (1982), ‘On the functional central limit theorem and the law of the iterated logarithm for Markov processes’, Z. Wahrsch. Verw. Gebiete 60, 185–201.Google Scholar
Binder, A., Simpson, G. and Lelièvre, T. (2015), ‘A generalized parallel replica dynamics’, J. Comput. Phys. 284, 595–616.Google Scholar
Blower, G. and Bolley, F. (2006), ‘Concentration of measure on product spaces with applications to Markov processes’, Studia Math. 175, 47–72.Google Scholar
Bobkov, S. and Götze, F. (1999), ‘Exponential integrability and transportation cost related to logarithmic Sobolev inequalities’, J. Funct. Anal. 163, 1–28.CrossRefGoogle Scholar
Bond, S. D. and Leimkuhler, B. J. (2007), Molecular dynamics and the accuracy of numerically computed averages. In Acta Numerica,Vol. 16, Cambridge University Press, pp. 1–65.Google Scholar
Bonetto, F., Lebowitz, J. L. and Rey-Bellet, L. (2000), Fourier’s law: A challenge to theorists. In Mathematical Physics 2000 (Fokas, A. et al. , ed.), Imperial College Press, pp. 128–150.Google Scholar
Bou-Rabee, N. and Hairer, M. (2013), ‘Nonasymptotic mixing of the MALA algorithm’, IMA J. Numer. Anal. 33, 80–110.Google Scholar
Bou-Rabee, N. and Owhadi, H. (2010), ‘Long-run accuracy of variational integrators in the stochastic context’, SIAM J. Numer. Anal. 48, 278–297.CrossRefGoogle Scholar
Bou-Rabee, N. and Vanden-Eijnden, E. (2012), ‘A patch that imparts unconditional stability to explicit integrators for Langevin-like equations’, J. Comput. Phys. 231, 2565–2580.Google Scholar
Bouchet, F. and Reygner, J. (2015), Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes. arXiv:1507.02104Google Scholar
Bovier, A. and den Hollander, F. (2015), Metastability: A Potential Theoretic Approach, Springer. To appear.CrossRefGoogle Scholar
Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2004a), ‘Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times’, J. Eur. Math. Soc. 6, 399–424.Google Scholar
Bovier, A., Gayrard, V. and Klein, M. (2004b), ‘Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues’, J. Eur. Math. Soc. 7, 69–99.Google Scholar
Bowman, G., Pande, V. and Noé, F. (2014), An Introduction to Markov State Models and their Application to Long Timescale Molecular Simulation, Springer.Google Scholar
Bréhier, C.-E., Gazeau, M., Goudenège, L., Lelièvre, T. and Rousset, M. (2016), ‘Unbiasedness of some generalized Adaptive Multilevel Splitting algorithms’, Ann. Appl. Probab., to appear.CrossRefGoogle Scholar
Brünger, A., Brooks, C. B. and Karplus, M. (1984), ‘Stochastic boundary conditions for molecular dynamics simulations of ST2 water’, Chem. Phys. Lett. 105, 495–500.Google Scholar
Buchete, N. and Hummer, G. (2008), ‘Peptide folding kinetics from replica exchange molecular dynamics’, Phys. Rev. E 77, 030902.Google Scholar
Bucklew, J. (2004), Introduction to Rare Event Simulation, Springer.Google Scholar
Burghelea, D. (1997), ‘Lectures on Witten–Helffer–Sjöstrand theory’, General Mathematics 5, 85–99.Google Scholar
Bussi, G. and Parrinello, M. (2007), ‘Accurate sampling using Langevin dynamics’, Phys. Rev. E 75, 056707.Google Scholar
Bussi, G., Laio, A. and Parrinello, M. (2006), ‘Equilibrium free energies from nonequilibrium metadynamics’, Phys. Rev. Lett. 96, 090601.CrossRefGoogle ScholarPubMed
Caflisch, R. (1998), Monte Carlo and quasi-Monte Carlo methods. In Acta Numerica,Vol. 7, Cambridge University Press, pp. 1–49.Google Scholar
Cancès, E., Defranceschi, M., Kutzelnigg, W., Le Bris, C. and Maday, Y. (2003), Computational quantum chemistry: A primer. In Handbook of Numerical Analysis (Ciarlet, P. G. and Le Bris, C., eds), Vol. X, Elsevier, pp. 3–270.Google Scholar
Carter, E., Ciccotti, G., Hynes, J. and Kapral, R. (1989), ‘Constrained reaction coordinate dynamics for the simulation of rare events’, Chem. Phys. Lett. 156, 472–477.Google Scholar
Cattiaux, P. (2014), Long time behavior of Markov processes. In Journées MAS 2012, Vol. 44 of ESAIM Proc., EDP Sciences, pp. 110–128.Google Scholar
Cattiaux, P., Guillin, A. and Wu, L.-M. (2009), ‘A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality’, Probab. Theory Rel. Fields 148, 285–304.Google Scholar
Cattiaux, P., Guillin, A. and Zitt, P.-A. (2013), ‘Poincaré inequalities and hitting times’, Ann. Inst. Henri Poincaré Probab. Statist. 49, 95–118.Google Scholar
Cérou, F. and Guyader, A. (2007), ‘Adaptive multilevel splitting for rare event analysis’, Stoch. Anal. Appl. 25, 417–443.CrossRefGoogle Scholar
Cérou, F., Guyader, A., Lelièvre, T. and Pommier, D. (2011), ‘A multiple replica approach to simulate reactive trajectories’, J. Chem. Phys. 134, 054108.Google Scholar
Chafaï, D. (2015), ‘From Boltzmann to random matrices and beyond’, Ann. Fac. Sci. Toulouse Math. 6, 641–689.Google Scholar
Chetrite, R. and Touchette, H. (2014), ‘Nonequilibrium Markov processes conditioned on large deviations’, Ann. Henri Poincaré 16, 2005–2057.CrossRefGoogle Scholar
Chipot, C. & Pohorille, A. (eds) (2007), Free Energy Calculations, Vol. 86 of Springer Series in Chemical Physics, Springer.Google Scholar
Chopin, N., Lelièvre, T. and Stoltz, G. (2012), ‘Free energy methods for Bayesian inference: Efficient exploration of univariate Gaussian mixture posteriors’, Statist. Comput. 22, 897–916.Google Scholar
Ciccotti, G. and Jacucci, G. (1975), ‘Direct computation of dynamical response by molecular-dynamics: The mobility of a charged Lennard-Jones particle’, Phys. Rev. Lett. 35, 789–792.Google Scholar
Ciccotti, G., Lelièvre, T. and Vanden-Eijnden, E. (2008), ‘Projection of diffusions on submanifolds: Application to mean force computation’, Commun. Pure Appl. Math. 61, 371–408.Google Scholar
Collet, P., Martínez, S. and San Martín, J. (2013), Quasi-Stationary Distributions, Springer.CrossRefGoogle Scholar
Cycon, H., Froese, R., Kirsch, W. and Simon, B. (1987), Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer.Google Scholar
Darve, E. and Porohille, A. (2001), ‘Calculating free energy using average forces’, J. Chem. Phys. 115, 9169–9183.Google Scholar
Dautray, R. and Lions, J.-L. (1990), Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3: Spectral Theory and Applications, Springer.Google Scholar
Dellago, C. and Bolhuis, P. (2009), Transition path sampling and other advanced simulation techniques for rare events. In Advanced Computer Simulation Approaches for Soft Matter Sciences III, Vol. 221 of Advances in Polymer Science, Springer, pp. 167–233.Google Scholar
Dellago, C., Bolhuis, P. and Chandler, D. (1999), ‘On the calculation of reaction rate constants in the transition path ensemble’, J. Chem. Phys 110, 6617–6625.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998), Large Deviations Techniques and Applications, second edition, Springer.Google Scholar
den Hollander, F. (2000), Large Deviations, Vol. 14 of Fields Institute Monographs, AMS.Google Scholar
den Otter, W. K. and Briels, W. J. (1998), ‘The calculation of free-energy differences by constrained molecular-dynamics simulations’, J. Chem. Phys. 109, 4139–4146.CrossRefGoogle Scholar
Derrida, B., Lebowitz, J. L. and Speer, E. R. (2002), ‘Large deviation of the density profile in the steady state of the open symmetric simple exclusion process’, J. Statist. Phys. 107, 599–634.CrossRefGoogle Scholar
Deuschel, J.-D. and Stroock, D. W. (1989), Large Deviations, Vol. 137 of Pure and Applied Mathematics, Academic.Google Scholar
Dhar, A. (2008), ‘Heat transport in low-dimensional systems’, Adv. Phys. 57, 457–537.Google Scholar
Dickson, B., Legoll, F., Lelièvre, T., Stoltz, G. and Fleurat-Lessard, P. (2010), ‘Free energy calculations: An efficient adaptive biasing potential method’, J. Phys. Chem. B 114, 5823–5830.CrossRefGoogle ScholarPubMed
Donsker, M. D. and Varadhan, S. (1975), ‘Asymptotic evaluation of certain Markov process expectations for large times I’, Commun. Pure Appl. Math. 28, 1–47.Google Scholar
Douc, R., Fort, G. and Guillin, A. (2009), ‘Subgeometric rates of convergence of $f$-ergodic strong Markov processes’, Stoch. Proc. Appl. 119, 897–923.CrossRefGoogle Scholar
Duane, S., Kennedy, A. D., Pendleton, B. J. and Roweth, D. (1987), ‘Hybrid Monte-Carlo’, Phys. Lett. B 195, 216–222.Google Scholar
Duncan, A. B., Lelièvre, T. and Pavliotis, G. A. (2016), ‘Variance reduction using nonreversible Langevin samplers’, J. Statist. Phys. 163, 457–491.Google Scholar
Dupuis, P. and Wang, H. (2004), ‘Importance sampling, large deviations, and differential games’, Stoch. and Stoch. Reports 76, 481–508.Google Scholar
Dupuis, P., Spiliopoulos, K. and Wang, H. (2012), ‘Importance sampling for multiscale diffusions’, SIAM J. Multiscale Model. Simul. 10, 1–27.Google Scholar
Dupuis, P., Spiliopoulos, K. and Zhou, X. (2015), ‘Escaping from an attractor: Importance sampling and rest points I’, Ann. Appl. Probab. 25, 2909–2958.Google Scholar
E, W. and Vanden-Eijnden, E. (2004), Metastability, conformation dynamics, and transition pathways in complex systems. In Multiscale Modelling and Simulation, Vol. 39 of Lecture Notes in Computational Science and Engineering, Springer, pp. 35–68.Google Scholar
E, W., Ren, W. and Vanden-Eijnden, E. (2002), ‘String method for the study of rare events’, Phys. Rev. B 66, 052301.Google Scholar
E, W., Ren, W. and Vanden-Eijnden, E. (2005), ‘Finite temperature string method for the study of rare events’, J. Phys. Chem. B 109, 6688–6693.Google Scholar
Eckmann, J.-P. and Hairer, M. (2003), ‘Spectral properties of hypoelliptic operators’, Commun. Math. Phys. 235, 233–253.Google Scholar
Español, P. and Warren, P. (1995), ‘Statistical-mechanics of dissipative particle dynamics’, Europhys. Lett. 30, 191–196.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986), Markov Processes: Probability and Mathematical Statistics, Wiley.Google Scholar
Evans, D. J. and Morriss, G. P. (2008), Statistical Mechanics of Nonequilibrium Liquids, Cambridge University Press.Google Scholar
Evans, L. and Gariepy, R. (1992), Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC.Google Scholar
Faou, E. (2012), Geometricx Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society.Google Scholar
Faradjian, A. and Elber, R. (2004), ‘Computing time scales from reaction coordinates by milestoning’, J. Chem. Phys. 120, 10880–10889.CrossRefGoogle ScholarPubMed
Fathi, M. and Stoltz, G. (2015), Improving dynamical properties of stabilized discretizations of overdamped Langevin dynamics. arXiv:1505.04905v1Google Scholar
Fathi, M., Homman, A.-A. and Stoltz, G. (2014), ‘Error analysis of the transport properties of Metropolized schemes’, ESAIM Proc. 48, 341–363.Google Scholar
Fischer, A., Waldhausen, S., Horenko, I., Meerbach, E. and Schütte, C. (2007), ‘Identification of biomolecular conformations from incomplete torsion angle observations by hidden Markov models’, J. Comp. Chem. 28, 2453–2464.Google Scholar
Fleming, W. (1977), ‘Exit probabilities and optimal stochastic control’, Appl. Math. Optim. 4, 329–346.Google Scholar
Fleming, W. and Soner, H. (2006), Controlled Markov Processes and Viscosity Solutions, Springer.Google Scholar
Fort, G., Jourdain, B., Kuhn, E., Lelièvre, T. and Stoltz, G. (2014), ‘Efficiency of the Wang–Landau algorithm: A simple test case’, Appl. Math. Res. Express 2, 275–311.Google Scholar
Fort, G., Jourdain, B., Kuhn, E., Lelièvre, T. and Stoltz, G. (2015a), ‘Convergence of the Wang–Landau algorithm’, Math. Comp. 84, 2297–2327.Google Scholar
Fort, G., Jourdain, B., Lelièvre, T. and Stoltz, G. (2015b), Self-healing umbrella sampling: Convergence and efficiency. Accepted for publication in Statist. Comput. arXiv:1410.2109Google Scholar
Freddolino, P. L., Harrison, C. B., Liu, Y. and Schulten, K. (2010), ‘Challenges in protein folding simulations’, Nature Physics 6, 751–758.Google Scholar
Freidlin, M. and Wentzell, A. (1984), Random Perturbations of Dynamical Systems, Springer.Google Scholar
Friedman, A. (1975), Stochastic Differential Equations and Applications, Vol. 1, Academic.Google Scholar
Gallavotti, G. and Cohen, E. G. D. (1995), ‘Dynamical ensembles in stationary states’, J. Statist. Phys. 80, 931–970.Google Scholar
Geyer, C. J. (1994), Estimating normalizing constants and reweighting mixtures in Markov chain Monte Carlo. Technical report 565, School of Statistics, University of Minnesota.Google Scholar
Giles, M. B. (2008), ‘Multilevel Monte Carlo path simulation’, Oper. Res. 56, 607–617.Google Scholar
Giles, M. B. (2015), Multilevel Monte Carlo methods. In Acta Numerica,Vol. 24, Cambridge University Press, pp. 259–328.Google Scholar
Givon, D., Kupferman, R. and Stuart, A. (2004), ‘Extracting macroscopic dynamics: Model problems and algorithms’, Nonlinearity 17, R55–R127.Google Scholar
Goodman, J. B. and Lin, K. K. (2009), ‘Coupling control variates for Markov chain Monte Carlo’, J. Comput. Phys. 228, 7127–7136.Google Scholar
Gross, L. (1975), ‘Logarithmic Sobolev inequalities’, Amer. J. Math. 97, 1061–1083.Google Scholar
Grunewald, N., Otto, F., Villani, C. and Westdickenberg, M. (2009), ‘A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit’, Ann. Inst. H. Poincaré Probab. Statist. 45, 302–351.Google Scholar
Guionnet, A. and Zegarlinski, B. (2003), Lectures on logarithmic Sobolev inequalities. In Séminaire de Probabilités XXXVI, Vol. 1801 of Lecture Notes in Mathematics, Springer, pp. 1–134.Google Scholar
Guyader, A. (2015). Personal communication.Google Scholar
Hairer, E., Lubich, C. and Wanner, G. (2003), Geometric numerical integration illustrated by the Störmer–Verlet method. In Acta Numerica,Vol. 12, Cambridge University Press, pp. 399–450.Google Scholar
Hairer, E., Lubich, C. and Wanner, G. (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Vol. 31 of Springer Series in Computational Mathematics, Springer.Google Scholar
Hairer, M. and Majda, A. J. (2010), ‘A simple framework to justify linear response theory’, Nonlinearity 23, 909–922.Google Scholar
Hairer, M. and Mattingly, J. C. (2011), Yet another look at Harris’ ergodic theorem for Markov chains. In Seminar on Stochastic Analysis, Random Fields and Applications VI, Vol. 63 of Progr. Probab., Birkhä user/Springer, pp. 109–117.Google Scholar
Hairer, M. and Pavliotis, G. (2008), ‘From ballistic to diffusive behavior in periodic potentials’, J. Statist. Phys. 131, 175–202.Google Scholar
Hartmann, C. and Schütte, C. (2012), ‘Efficient rare event simulation by optimal nonequilibrium forcing’, J. Statist. Mech. Theory Exp. 2012, P11004.Google Scholar
Hartmann, C., Schütte, C., Weber, M. and Zhang, W. (2015), Importance sampling in path space for diffusion processes. arXiv:1502.07899Google Scholar
Hastings, W. K. (1970), ‘Monte Carlo sampling methods using Markov chains and their applications’, Biometrika 57, 97–109.Google Scholar
Helffer, B. and Nier, F. (2005), Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians, Vol. 1862 of Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Helffer, B. and Nier, F. (2006), ‘Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: The case with boundary’, Mémoires de la SMF 105.Google Scholar
Helffer, B. and Sjöstrand, J. (1984), ‘Multiple wells in the semiclassical limit I’, Comm. Partial Diff. Equations 9, 337–408.Google Scholar
Helffer, B. and Sjöstrand, J. (1985), ‘Puits multiples en mécanique semi-classique IV: Etude du complexe de Witten’, Comm. Partial Diff. Equations 10, 245–340.Google Scholar
Helffer, B., Klein, M. and Nier, F. (2004), ‘Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach’, Mat. Contemp. 26, 41–85.Google Scholar
Henderson, S. G. (1997), Variance reduction via an approximating Markov process. PhD thesis, Department of Operations Research, Stanford University.Google Scholar
Hénin, J. and Chipot, C. (2004), ‘Overcoming free energy barriers using unconstrained molecular dynamics simulations’, J. Chem. Phys. 121, 2904–2914.Google Scholar
Henkelman, G. and Jónsson, H. (1999), ‘A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives’, J. Chem. Phys. 111, 7010–7022.Google Scholar
Hérau, F. and Nier, F. (2004), ‘Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with a high-degree potential’, Arch. Rational Mech. Anal. 171, 151–218.Google Scholar
Hérau, F., Hitrik, M. and Sjöstrand, J. (2011), ‘Tunnel effect and symmetries for Kramers–Fokker–Planck type operators’, J. Inst. Math. Jussieu 10, 567–634.Google Scholar
Hohenberg, P. and Kohn, W. (1964), ‘Inhomogeneous electron gas’, Phys. Rev. B 136, 864–871.Google Scholar
Holley, R. and Stroock, D. (1987), ‘Logarithmic Sobolev inequalities and stochastic Ising models’, J. Statist. Phys. 46, 1159–1194.Google Scholar
Hoogerbrugge, P. J. and Koelman, J. M. V. A. (1992), ‘Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics’, Europhys. Lett. 19, 155–160.CrossRefGoogle Scholar
Hoover, W. G. (1985), ‘Canonical dynamics: Equilibrium phase-space distributions’, Phys. Rev. A 31, 1695–1697.Google Scholar
Hörmander, L. (1967), ‘Hypoelliptic second order differential equations’, Acta Math. 119, 147–171.Google Scholar
Hörmander, L. (1985), The Analysis of Linear Partial Differential Operators I–IV, Springer.Google Scholar
Hulse, R. J., Howley, R. and Wilding, W. (2005), ‘Transient nonequilibrium molecular dynamic simulations of thermal conductivity I: Simple fluids’, Int. J. Thermophys. 26, 1–12.Google Scholar
Hummer, G. (2004), ‘From transition paths to transition states and rate coefficients’, J. Chem. Phys. 120, 516–523.Google Scholar
Hummer, G. and Szabo, A. (2001), ‘Free energy reconstruction from nonequilibrium single-molecule pulling experiments’, Proc. Nat. Acad. Sci. 98, 3658–3661.Google Scholar
Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S. J. (1993), ‘Accelerating Gaussian diffusions’, Ann. Appl. Probab. 3, 897–913.Google Scholar
Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S. J. (2005), ‘Accelerating diffusions’, Ann. Appl. Probab. 15, 1433–1444.Google Scholar
Izaguirre, J. A. and Hampton, S. S. (2004), ‘Shadow hybrid Monte Carlo: An efficient propagator in phase space of macromolecules’, J. Chem. Phys. 200, 581–604.Google Scholar
Jack, R. and Sollich, P. (2010), ‘Large deviations and ensembles of trajectories in stochastic models’, Prog. Theor. Phys. Supp. 184, 304–317.Google Scholar
Jarzynski, C. (1997), ‘Equilibrium free energy differences from nonequilibrium measurements: A master equation approach’, Phys. Rev. E 56, 5018–5035.Google Scholar
Jerrum, M., Son, J.-B., Tetali, P. and Vigoda, E. (2004), ‘Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains’, Ann. Appl. Probab. 14, 1741–1765.Google Scholar
Jónsson, H., Mills, G. and Jacobsen, K. (1998), Nudged elastic band method for finding minimum energy paths of transitions. In Classical and Quantum Dynamics in Condensed Phase Simulations (Berne, B. J., Ciccotti, G. and Coker, D. F., eds), World Scientific, pp. 385–404.CrossRefGoogle Scholar
Jorgensen, W. L. and Tirado-Rives, J. (2005), ‘Potential energy functions for atomic-level simulations of water and organic and biomolecular systems’, Proc. Natl Acad. Sci. USA 102, 6665–6670.Google Scholar
Joubaud, R., Pavliotis, G. and Stoltz, G. (2015), ‘Langevin dynamics with space-time periodic nonequilibrium forcing’, J. Statist. Phys. 158, 1–36.Google Scholar
Jourdain, B., Lelièvre, T. and Roux, R. (2010), ‘Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process’, ESAIM-Math. Model. Num. 44, 831–865.Google Scholar
Kevrekidis, I. G. and Samaey, G. (2009), ‘Equation-free multiscale computation: Algorithms and applications’, Annu. Rev. Phys. Chem. 60, 321–344.Google Scholar
Khasminskii, R. (2012), Stochastic Stability of Differential Equations, Vol. 66 of Stochastic Modelling and Applied Probability, second edition, Springer.Google Scholar
Kipnis, C. and Landim, C. (1999), Scaling Limits of Interacting Particle Systems, Springer.Google Scholar
Kirkwood, J. (1935), ‘Statistical mechanics of fluid mixtures’, J. Chem. Phys. 3, 300–313.Google Scholar
Kliemann, W. (1987), ‘Recurrence and invariant measures for degenerate diffusions’, Ann. Probab. 15, 690–707.Google Scholar
Kloeden, P. E. and Platen, E. (1992), Numerical Solution of Stochastic Differential Equations, Vol. 23 of Applications of Mathematics, Springer.Google Scholar
Kohn, W. and Sham, L. J. (1965), ‘Self-consistent equations including exchange and correlation effects’, Phys. Rev. 140, A1133–A1138.Google Scholar
Komorowski, T., Landim, C. and Olla, S. (2012), Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation, Vol. 345 of Grundlehren der Mathematischen Wissenschaften, Springer.CrossRefGoogle Scholar
Kong, A., Liu, J. S. and Wong, W. H. (1994), ‘Sequential imputation and Bayesian missing data problems’, J. Amer. Statist. Assoc. 89, 278–288.Google Scholar
Kong, A., McCullagh, P., Meng, X. L., Nicolae, D. and Tan, Z. (2003), ‘A theory of statistical models for Monte-Carlo integration’, J. Roy. Statist. Soc. B 65, 585–618.Google Scholar
Kopec, M. (2015), ‘Weak backward error analysis for Langevin process’, BIT 55, 1057–1103.Google Scholar
Kopec, M. (2015), ‘Weak backward error analysis for overdamped Langevin processes’, IMA J. Numer. Anal. 35, 583–614.Google Scholar
Kum, O., Dickson, B., Stuart, S., Uberuaga, B. and Voter, A. (2004), ‘Parallel replica dynamics with a heterogeneous distribution of barriers: Application to $n$-hexadecane pyrolysis’, J. Chem. Phys. 121, 9808–9819.Google Scholar
Kundu, A., Sabhapandit, S. and Dhar, A. (2011), ‘Application of importance sampling to the computation of large deviations in nonequilibrium processes’, Phys. Rev. E 83, 031119.CrossRefGoogle Scholar
Kurchan, J. (1998), ‘Fluctuation theorem for stochastic dynamics’, J. Phys. A 31, 3719–3729.Google Scholar
Laio, A. and Parrinello, M. (2002), ‘Escaping free-energy minima’, Proc. Natl Acad. Sci. USA 99, 12562–12566.Google Scholar
Lamberton, D. and Pagès, G. (2002), ‘Recursive computation of the invariant distribution of a diffusion’, Bernoulli 8, 367–405.Google Scholar
Le Bris, C. (2005), Computational chemistry from the perspective of numerical analysis. In Acta Numerica,Vol. 14, Cambridge University Press, pp. 363–444.Google Scholar
Le Bris, C. and Lions, P.-L. (2008), ‘Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients’, Comm. Partial Diff. Equations 33, 1272–1317.Google Scholar
Le Bris, C., Lelièvre, T., Luskin, M. and Perez, D. (2012), ‘A mathematical formalization of the parallel replica dynamics’, Monte Carlo Methods Appl. 18, 119–146.Google Scholar
Le Peutrec, D., Nier, F. and Viterbo, C. (2013), ‘Precise Arrhenius law for $p$-forms: The Witten Laplacian and Morse–Barannikov complex’, Ann. Henri Poincaré 14, 567–610.Google Scholar
Lebowitz, J. L. and Spohn, H. (1999), ‘A Gallavotti–Cohen-type symmetry in the large deviation functional for stochastic dynamics’, J. Statist. Phys. 95, 333–365.Google Scholar
Ledoux, M. (2001), Logarithmic Sobolev inequalities for unbounded spin systems revisited. In Séminaire de Probabilités XXXV, Vol. 1755 of Lecture Notes in Mathematics, Springer, pp. 167–194.Google Scholar
Lee, E. H., Hsin, J., Sotomayor, M., Comellas, G. and Schulten, K.v (2009), ‘Discovery through the computational microscope’, Structure 17, 1295–1306.Google Scholar
Legoll, F. and Lelièvre, T. (2010), ‘Effective dynamics using conditional expectations’, Nonlinearity 23, 2131–2163.Google Scholar
Leimkuhler, B. and Matthews, C. (2013a), ‘Rational construction of stochastic numerical methods for molecular sampling’, Appl. Math. Res. Express 34–56.Google Scholar
Leimkuhler, B. and Matthews, C. (2013b), ‘Robust and efficient configurational molecular sampling via Langevin dynamics’, J. Chem. Phys. 138, 174102.Google Scholar
Leimkuhler, B. and Matthews, C. (2015), Molecular Dynamics: With Deterministic and Stochastic Numerical Methods, Springer.Google Scholar
Leimkuhler, B. and Reich, S. (2004), Simulating Hamiltonian Dynamics, Cambridge University Press.Google Scholar
Leimkuhler, B., Matthews, C. and Stoltz, G. (2015), ‘The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics’, IMA J. Numer. Anal., in press.Google Scholar
Leimkuhler, B., Noorizadeh, E. and Theil, F. (2009), ‘A gentle stochastic thermostat for molecular dynamics’, J. Statist. Phys. 135, 261–277.Google Scholar
Lelièvre, T. (2009), ‘A general two-scale criteria for logarithmic Sobolev inequalities’, J. Funct. Anal. 256, 2211–2221.Google Scholar
Lelièvre, T. (2013), Two mathematical tools to analyze metastable stochastic processes. In Numerical Mathematics and Advanced Applications 2011 (Cangiani, A., Davidchack, R. L., Georgoulis, E., Gorban, A. N., Levesley, J. and Tretyakov, M. V., eds), Springer, pp. 791–810.Google Scholar
Lelièvre, T. and Minoukadeh, K. (2011), ‘Long-time convergence of an adaptive biasing force method: The bi-channel case’, Arch. Rational Mech. Anal. 202, 1–34.Google Scholar
Lelièvre, T. and Nier, F. (2015), ‘Low temperature asymptotics for quasi-stationary distributions in a bounded domain’, Analysis & PDE 8, 561–628.Google Scholar
Lelièvre, T., Nier, F. and Pavliotis, G. A. (2013), ‘Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion’, J. Statist. Phys. 152, 237–274.Google Scholar
, Lelièvre, T., Rousset, M. and Stoltz, G. (2007), ‘Computation of free energy profiles with adaptive parallel dynamics’, J. Chem. Phys. 126, 134111.Google Scholar
Lelièvre, T., Rousset, M. and Stoltz, G. (2008), ‘Long-time convergence of an adaptive biasing force method’, Nonlinearity 21, 1155–1181.Google Scholar
Lelièvre, T., Rousset, M. and Stoltz, G. (2010), Free Energy Computations: A Mathematical Perspective, Imperial College Press.Google Scholar
Lelièvre, T., Rousset, M. and Stoltz, G. (2012), ‘Langevin dynamics with constraints and computation of free energy differences’, Math. Comp. 81, 2071–2125.Google Scholar
Lepri, S., Livi, R. and Politi, A. (2003), ‘Thermal conduction in classical low-dimensional lattices’, Phys. Rep. 377, 1–80.Google Scholar
Lieb, E. H. (1983), ‘Density functional theory for Coulomb systems’, Int. J. Quant. Chem. 24, 243–277.Google Scholar
Lipparini, F., Lagardère, L., Raynaud, C., Stamm, B., Cancès, E., Mennucci, B., Schnieders, M., Ren, P., Maday, Y. and Piquemal, J.-P. (2015), ‘Polarizable molecular dynamics in a polarizable continuum solvent’, J. Chem. Theory Comput. 11, 623–634.Google Scholar
Lu, J. and Nolen, J. (2015), ‘Reactive trajectories and the transition path process’, Probab. Theory Rel. Fields 161, 195–244.Google Scholar
Luskin, M. and Ortner, C. (2013), Atomistic-to-continuum coupling. In Acta Numerica,Vol. 22, Cambridge University Press, pp. 397–508.Google Scholar
Madras, N. and Randall, D. (2002), ‘Markov chain decomposition for convergence rate analysis’, Ann. Appl. Probab. 12, 581–606.Google Scholar
Maier, R. S. and Stein, D. L. (1993), ‘Escape problem for irreversible systems’, Phys. Rev. E 48, 931–938.Google Scholar
Maier, R. S. and Stein, D. L. (1997), ‘Limiting exit location distributions in the stochastic exit problem’, SIAM J. Appl. Math. 57, 752–790.Google Scholar
Mao, X. (2008), Stochastic Differential Equations and Applications, second edition, Horwood, Chichester.Google Scholar
Maragliano, L., Vanden-Eijnden, E. and Roux, B. (2009), ‘Free energy and kinetics of conformational transitions from Voronoi tessellated milestoning with restraining potentials’, J. Chem. Theory Comput. 5, 2589–2594.Google Scholar
Markowich, P. A. and Villani, C. (2000), ‘On the trend to equilibrium for the Fokker–Planck equation: An interplay between physics and functional analysis’, Mat. Contemp. 19, 1–29.Google Scholar
Marsden, J. E. and West, M. (2001), Discrete mechanics and variational integrators. In Acta Numerica,Vol. 10, Cambridge University Press, pp. 357–514.Google Scholar
Marsili, S., Barducci, A., Chelli, R., Procacci, P. and Schettino, V. (2006), ‘Self-healing Umbrella Sampling: A non-equilibrium approach for quantitative free energy calculations’, J. Phys. Chem. B 110, 14011–14013.Google Scholar
Matkowsky, B. J. and Schuss, Z. (1979), ‘The exit problem: A new approach to diffusion across potential barriers’, SIAM J. Appl. Math. 36, 604–623.Google Scholar
Mattingly, J. C., Stuart, A. M. and Higham, D. J. (2002), ‘Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise’, Stoch. Proc. Appl. 101, 185–232.Google Scholar
, Melchionna, S. (2007), ‘Design of quasisymplectic propagators for Langevin dynamics’, J. Chem. Phys. 127, 044108.Google Scholar
Meng, X.-L. and Wong, W. H. (1996), ‘Simulating ratios of normalizing constants via a simple identity: A theoretical exploration’, Statist. Sinica 6, 831–860.Google Scholar
Menz, G. and Schlichting, A. (2014), ‘Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape’, Ann. Probab. 42, 1809–1884.Google Scholar
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953), ‘Equations of state calculations by fast computing machines’, J. Chem. Phys. 21, 1087–1091.Google Scholar
Meyn, S. P. and Tweedie, R. L. (2009), Markov Chains and Stochastic Stability, second edition, Cambridge University Press.Google Scholar
Milstein, G. N. and Tretyakov, M. V. (2004), Stochastic Numerics for Mathematical Physics, Scientific Computation, Springer.Google Scholar
Minoukadeh, K., Chipot, C. and Lelièvre, T. (2010), ‘Potential of mean force calculations: a multiple-walker adaptive biasing force approach’, J. Chem. Theory Comput. 6, 1008–1017.Google Scholar
Miron, R. and Fichthorn, K. (2003), ‘Accelerated molecular dynamics with the bond–boost method’, J. Chem. Phys. 119, 6210–6216.Google Scholar
Mori, H. (1965), ‘Transport, collective motion, and Brownian motion’, Prog. Theor. Phys. 33, 423–450.Google Scholar
Mousseau, N. and Barkema, G. (1998), ‘Traveling through potential energy landscapes of disordered materials: The activation–relaxation technique’, Phys. Rev. E 57, 2419–2424.Google Scholar
Naeh, T., Klosek, M. M., Matkowsky, B. J. and Schuss, Z. (1990), ‘A direct approach to the exit problem’, SIAM J. Appl. Math. 50, 595–627.Google Scholar
Nier, F. (2014), Boundary conditions and subelliptic estimates for geometric Kramers–Fokker–Planck operators on manifolds with boundaries. arXiv:1309.5070Google Scholar
Nosé, S. (1984a), ‘A molecular-dynamics method for simulations in the canonical ensemble’, Mol. Phys. 52, 255–268.Google Scholar
Nosé, S. (1984b), ‘A unified formulation of the constant temperature molecular-dynamics methods’, J. Chem. Phys. 81, 511–519.Google Scholar
Nummelin, E. (1984), General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press.Google Scholar
Olivieri, E. and Vares, M. (2005), Large Deviations and Metastability, Cambridge University Press.Google Scholar
Olla, S. and Letizia, V. (2015), Non-equilibrium isothermal transformations in a temperature gradient from a microscopic dynamics. arXiv:1505.05002v1Google Scholar
Otto, F. and Reznikoff, M. (2007), ‘A new criterion for the logarithmic Sobolev inequality and two applications’, J. Funct. Anal. 243, 121–157.Google Scholar
Otto, F. and Villani, C. (2000), ‘Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality’, J. Funct. Anal. 173, 361–400.Google Scholar
Pardoux, E. and Veretennikov, A. Y. (2005), ‘On the Poisson equation and diffusion approximation 3’, Ann. Probab. 33, 1111–1133.Google Scholar
Piana, S. and Laio, A. (2007), ‘A bias-exchange approach to protein folding’, J. Phys. Chem. B 111, 4553–4559.Google Scholar
Platen, E. (1999), An introduction to numerical methods for stochastic differential equations. In Acta Numerica,Vol. 8, Cambridge University Press, pp. 197–246.Google Scholar
Redon, S., Stoltz, G. and Trstanova, Z. (2016), Error analysis of modified Langevin dynamics. arXiv:1601.07411Google Scholar
Reed, M. and Simon, B. (1975a), Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic.Google Scholar
Reed, M. and Simon, B. (1975b), Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators, Academic.Google Scholar
Reimann, P. (2002), ‘Brownian motors: Noisy transport far from equilibrium’, Phys. Rep. 61, 57–265.Google Scholar
Revuz, D. and Yor, M. (1999), Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften, Springer.Google Scholar
Rey-Bellet, L. (2006), Ergodic properties of Markov processes. In vol. 1881 of Lecture Notes in Mathematics (Attal, S., Joye, A. and Pillet, C.-A., eds), Springer, pp. 1–39.Google Scholar
Rey-Bellet, L. and Spiliopoulos, K. (2015), ‘Irreversible Langevin samplers and variance reduction: A large deviations approach’, Nonlinearity 28, 2081–2103.Google Scholar
Rey-Bellet, L. and Spiliopoulos, K. (2016), Improving the convergence of reversible samplers. arXiv:1601.08118Google Scholar
Roberts, G. O. and Rosenthal, J. S. (1998), ‘Optimal scaling of discrete approximations to Langevin diffusions’, J. Roy. Statist. Soc. B 60, 255–268.Google Scholar
Roberts, G. O., Gelman, A. and Gilks, W. R. (1997), ‘Weak convergence and optimal scaling of random walk Metropolis algorithms’, Ann. Appl. Probab. 7, 110–120.Google Scholar
Rohrdanz, M. A., Zheng, W., Maggioni, M. and Clementi, C. (2011), ‘Determination of reaction coordinates via locally scaled diffusion map’, J. Chem. Phys. 134, 124116.Google Scholar
Rossky, P. J., Doll, J. D. and Friedman, H. L. (1978), ‘Brownian dynamics as smart Monte Carlo simulation’, J. Chem. Phys. 69, 4628–4633.Google Scholar
Royer, G. (2007), An Initiation to Logarithmic Sobolev Inequalities, AMS.Google Scholar
Rudin, W. (1987), Real and Complex Analysis, McGraw-Hill.Google Scholar
Rupp, M. (2015), ‘Preface to special issue on machine learning and quantum mechanics’, Int. J. Quantum Chem. 115, 1003–1004.Google Scholar
Samanta, A. and E, W. (2012), ‘Atomistic simulations of rare events using gentlest ascent dynamics’, J. Chem. Phys. 136, 124104.Google Scholar
Sarich, M. and Schütte, C. (2013), Metastability and Markov state Models in Molecular Dynamics, Vol. 24 of Courant Lecture Notes, AMS.Google Scholar
Sarich, M., Noé, F. and Schütte, C. (2010), ‘On the approximation quality of Markov state models’, Multiscale Model. Simul. 8, 1154–1177.Google Scholar
Schlick, T. (2002), Molecular Modeling and Simulation, Springer.Google Scholar
Schütte, C. (1998), Conformational dynamics: Modelling, theory, algorithm and application to biomolecules. Habilitation dissertation, Freie Universität Berlin.Google Scholar
Schütte, C., Noé, F., Lu, J., Sarich, M. and Vanden-Eijnden, E. (2011), ‘Markov state models based on milestoning’, J. Chem. Phys. 134, 204105.Google Scholar
Schütte, C., Winkelmann, S. and Hartmann, C. (2012), ‘Optimal control of molecular dynamics using Markov state models’, Math. Program. 134, 259–282.Google Scholar
Shardlow, T. and Yan, Y. B. (2006), ‘Geometric ergodicity for dissipative particle dynamics’, Stoch. Dynam. 6, 123–154.Google Scholar
Shirts, M. and Chodera, J. (2008), ‘Statistically optimal analysis of samples from multiple equilibrium states’, J. Chem. Phys. 124, 124105.Google Scholar
Siegmund, D. (1976), ‘Importance sampling in the Monte Carlo study of sequential tests’, Ann. Stat. 4, 673–684.Google Scholar
Simon, B. (1984), ‘Semiclassical analysis of low lying eigenvalues II: Tunneling’, Ann. of Math. 120, 89–118.Google Scholar
Skeel, R. D. and Izaguirre, J. A. (2002), ‘An impulse integrator for Langevin dynamics’, Mol. Phys. 100, 3885–3891.Google Scholar
Sorensen, M. and Voter, A. (2000), ‘Temperature-accelerated dynamics for simulation of infrequent events’, J. Chem. Phys. 112, 9599–9606.Google Scholar
Sprik, M. and Ciccoti, G. (1998), ‘Free energy from constrained molecular dynamics’, J. Chem. Phys. 109, 7737–7744.Google Scholar
Talay, D. (2002), ‘Stochastic Hamiltonian dissipative systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme’, Markov Proc. Rel. Fields 8, 163–198.Google Scholar
Talay, D. and Tubaro, L. (1990), ‘Expansion of the global error for numerical schemes solving stochastic differential equations’, Stoch. Proc. Appl. 8, 94–120.Google Scholar
Tan, Z. (2004), ‘On a likelihood approach for Monte-Carlo integration’, J. Am. Stat. Assoc. 99, 1027–1036.Google Scholar
Tersoff, J. (1989), ‘Modeling solid-state chemistry: Interatomic potentials for multicomponent systems’, Phys. Rev. B 39, 5566–5568.Google Scholar
Thalmann, F. and Farago, J. (2007), ‘Trotter derivation of algorithms for Brownian and dissipative particle dynamics’, J. Chem. Phys. 127, 124109.Google Scholar
Tierney, L. (1994), ‘Markov chains for exploring posterior distributions’, Ann. Stat. 22, 1701–1762.Google Scholar
Tuckerman, M. (2010), Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press.Google Scholar
Vaikuntanathan, S. and Jarzynski, C. (2008), ‘Escorted free energy simulations: Improving convergence by reducing dissipation’, Phys. Rev. Lett. 100, 190601.Google Scholar
van Duin, A. C. T., Dasgupta, S., Lorant, F. and Goddard, W. A. III (2001), ‘ReaxFF: A reactive force field for hydrocarbons’, J. Phys. Chem. A 105, 9396–9409.Google Scholar
van Erp, T. and Bolhuis, P. (2005), ‘Elaborating transition interface sampling methods’, J. Comp. Phys. 205, 157–181.Google Scholar
van Erp, T., Moroni, D. and Bolhuis, P. (2003), ‘A novel path sampling method for the calculation of rate constants’, J. Chem. Phys. 118, 7762–7774.Google Scholar
Vanden-Eijnden, E. and Weare, J. (2012), ‘Rare event simulation of small noise diffusions’, Comm. Pure Appl. Math. 65, 1770–1803.Google Scholar
Vanden-Eijnden, E., Venturoli, M., Ciccotti, G. and Elber, R (2008), ‘On the assumptions underlying milestoning’, J. Chem. Phys. 129, 174102.Google Scholar
Verlet, L. (1967), ‘Computer “experiments” on classical fluids I: Thermodynamical properties of Lennard–Jones molecules’, Phys. Rev. 159, 98–103.Google Scholar
Villani, C. (2002), A review of mathematical topics in collisional kinetic theory. In Handbook of Mathematical Fluid Dynamics,Vol. I, North-Holland, pp. 71–305.Google Scholar
Villani, C. (2003), Topics in Optimal Transportation, Vol. 58 of Graduate Studies in Mathematics, AMS.Google Scholar
Villani, C. (2009), Hypocoercivity, Vol. 950 of Memoirs of the American Mathematical Society, AMS.Google Scholar
Voter, A. F. (1997), ‘A method for accelerating the molecular dynamics simulation of infrequent events’, J. Chem. Phys. 106, 4665–4677.Google Scholar
Voter, A. F. (1998), ‘Parallel replica method for dynamics of infrequent events’, Phys. Rev. B 57, R13 985.Google Scholar
Voter, A. F. (2007), Introduction to the Kinetic Monte Carlo method. In Radiation Effects in Solids, (Sickafus, K. E., Kotomin, E. A. and Uberuaga, B. P., eds), Vol. 235 of NATO Science Series, Springer, pp. 1–23.Google Scholar
Voter, A. F., Montalenti, F. and Germann, T. C. (2002), ‘Extending the time scale in atomistic simulation of materials’, Ann. Rev. Mater. Res. 32, 321–346.Google Scholar
Wales, D. (2003), Energy Landscapes, Cambridge University Press.Google Scholar
Wallqvist, A. and Mountain, R. D. (1999), ‘Molecular models of water: Derivation and description’, Rev. Comput. Chem. 13, 183–247.Google Scholar
Wang, F. and Landau, D. (2001a), ‘Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram’, Phys. Rev. E 64, 056101.Google Scholar
Wang, F. and Landau, D. (2001b), ‘Efficient, multiple-range random walk algorithm to calculate the density of states’, Phys. Rev. Lett. 86, 2050–2053.Google Scholar
Widom, B. (1963), ‘Some topics in the theory of fluids’, J. Chem. Phys. 39, 2808–2812.Google Scholar
Witten, E. (1982), ‘Supersymmetry and Morse theory’, J. Differential Geom. 17, 661–692.Google Scholar
Zhang, J. and Du, Q. (2012), ‘Shrinking dimer dynamics and its applications to saddle point search’, SIAM J. Numer. Anal. 50, 1899–1921.Google Scholar
Zhang, W. (2001), Lectures on Chern–Weil Theory and Witten Deformations, Vol. 4 of Nankai Tracts in Mathematics, World Scientific.Google Scholar
Zhang, W., Wang, H., Hartmann, C., Weber, M. and Schütte, C. (2014), ‘Applications of the cross-entropy method to importance sampling and optimal control of diffusions’, SIAM J. Sci. Comput. 36, A2654–A2672.Google Scholar
Zhao, R., Shen, J. and Skeel, R. (2010), ‘Maximum flux transition paths of conformational change’, J. Chem. Theory Comput. 6, 2411–2423.Google Scholar
Zwanzig, R. (1954), ‘High-temperature equation of state by a perturbation method: I. nonpolar gases’, J. Chem. Phys. 22, 1420–1426.Google Scholar
Zwanzig, R. (1973), ‘Nonlinear generalized Langevin equations’, J. Stat. Phys. 9, 215–220.Google Scholar