The adaptive biasing force algorithm with non-conservative forces and related topics

Lelièvre, Tony; Maurin, Lise; Monmarché, Pierre

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 2, pp. 529-564

We propose a study of the Adaptive Biasing Force method’s robustness under generic (possibly non-conservative) forces. We first ensure the flat histogram property is satisfied in all cases. We then introduce a fixed point problem yielding the existence of a stationary state for both the Adaptive Biasing Force and Projected Adapted Biasing Force algorithms, relying on generic bounds on the invariant probability measures of homogeneous diffusions. Using classical entropy techniques, we prove the exponential convergence of both biasing force and law as time goes to infinity, for both the Adaptive Biasing Force and the Projected Adaptive Biasing Force methods.

Classification : 35B40, 60J60
Keywords: Adaptive Bisaing Force, nonlinear Fokker-Planck equation, long-time behaviour, free energy, entropy techniques

[1] H. Alrachid and T. Lelièvre, Long-time convergence of an adaptive biasing force method: variance reduction by Helmholtz projection. J. Comput. Math. 1 (2015) 55–82. | MR | Zbl

[2] L. Ambrosio, A. Carlotto and A. Massaccesi, Lectures on Elliptic Partial Differential Equations (2018). | MR

[3] D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion operators. Vol. 348 of Grundlehren der mathematischen Wissenschaften. Springer (2014). | MR | Zbl

[4] V. I. Bogachev, N. V. Krylov, M. Röckner and S. V. Shaposhnikov, Fokker–Planck–Kolmogorov equations. American Mathematical Society (2015). | MR

[5] H. Brézis, Analyse fonctionnelle : théorie et applications. Masson (1987). | Zbl | MR

[6] S. Chmiela, A. Tkatchenko, H. E. Sauceda, I. Poltavsky, K. Schütt and K.-R. Müller, Machine learning of accurate energy-conserving molecular force fields. Sci. Adv. 3 (2017) 05.

[7] E. Darve and A. Pohorille, Calculating free energies using average force. J. Chem. Phys. 115 (2001) 9169–9183.

[8] E. B. Dynkin, Markov Processes Volume I. Springer Verlag (1965). | Zbl

[9] E. B. Dynkin, Markov Processes, Volume II. Springer Verlag (1965). | MR | Zbl

[10] L. C. Evans, Partial Differential Equations. American Mathematical Society (2010). | MR | Zbl

[11] H. Fu, X. Shao, C. Chipot and W. Cai, Extended adaptive biasing force algorithm an on-the-fly implementation for accurate free-energy calculations. J. Chem. Theory Comput. 12 (2016) 3506–3513.

[12] P. Gkeka, G. Stoltz, A. Barati Farimani, Z. Belkacemi, M. Ceriotti, J. Chodera, A. R. Dinner, A. Ferguson, J.-B. Maillet, H. Minoux, C. Peter, F. Pietrucci, A. Silveira, A. Tkatchenko, Z. Trstanova, R. Wiewiora and T. Lelièvre, Machine learning force fields and coarse-grained variables in molecular dynamics: application to materials and biological systems. Preprint (2020) | arXiv

[13] J. Hénin and C. Chipot, Overcoming free energy barriers using unconstrained molecular dynamics simulations. J. Chem. Phys. 121 (2004) 2904–2914.

[14] L. Hörmander, Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147–171. | MR | Zbl

[15] B. Jourdain, T. Lelièvre and R. Roux, Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process. ESAIM: M2AN 44 (2010) 831–865. | MR | Zbl | Numdam

[16] X. Kong and C. L. Brooks, $\lambda$-dynamics: A new approach to free energy calculations. J. Chem. Phys. 105 (1996) 2414–2423.

[17] N. Krylov, On diffusion processes with drift in $L_d$. Probab. Theory Relat. Fields (2020). | MR | Zbl

[18] T. Lelièvre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics. Acta Numer. 25 (2016). | MR | Zbl

[19] T. Lelièvre, M. Rousset and G. Stoltz, Long-time convergence of an Adaptive Biasing Force method. Nonlinearity 21 (2008). | MR | Zbl

[20] T. Lelièvre, M. Rousset and G. Stoltz, Free Energy Computations. Imperial College Press (2010). | MR | Zbl

[21] G. Menz and A. Schlichting, Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape. Ann. Probab. 42 (2014) 1809–1884. | MR | Zbl

[22] S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability. Springer Verlag, London (1993). | MR | Zbl

[23] P. Monmarché, Generalized $\Gamma$ calculus and application to interacting particles on a graph. Potential Anal. 50 (2019) 439–466. | MR | Zbl

[24] A. Niklasson, C. Tymczak and M. Challacombe, Time-reversible born-oppenheimer molecular dynamics. Phys. Rev. Lett. 97 (2006) 10.

[25] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361–400. | MR | Zbl

[26] P. Pulay and G. Fogarasi, Fock matrix dynamics. Chem. Phys. Lett. 386 (2004) 03.