Continuum limit and preconditioned Langevin sampling of the path integral molecular dynamics

We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. We propose two preconditioned Langevin sampling dynamics, which are shown to have improved stability and sampling accuracy. We present a careful mode analysis of the preconditioned dynamics and show their connections to the normal mode, the staging coordinate and the Matsubara mode representation for ring polymers. In the case where the potential is quadratic, we show that the continuum limit of the preconditioned mass modified Langevin dynamics converges to its equilibrium exponentially fast, which suggests that the finite-dimensional counterpart has a dimension-independent convergence rate. In addition, the preconditioning techniques can be naturally applied to the multi-level quantum systems in the nonadiabatic regime, which are compatible with various numerical approaches.

[1]  G. Pavliotis.  Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations , 2014 .

[2]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[3]  John C Tully,et al.  Path-integral simulations beyond the adiabatic approximation. , 2007, The Journal of chemical physics.

[4]  M. Thoss,et al.  Semiclassical Description of Nonadiabatic Quantum Dynamics , 1997 .

[5]  Michele Parrinello,et al.  Efficient stochastic thermostatting of path integral molecular dynamics. , 2010, The Journal of chemical physics.

[6]  Raymond Kapral,et al.  Progress in the theory of mixed quantum-classical dynamics. , 2006, Annual review of physical chemistry.

[7]  Bruce J. Berne,et al.  On the Simulation of Quantum Systems: Path Integral Methods , 1986 .

[8]  Thomas F. Miller,et al.  Ring-polymer molecular dynamics: quantum effects in chemical dynamics from classical trajectories in an extended phase space. , 2013, Annual review of physical chemistry.

[9]  A. Stuart,et al.  ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART II: THE NONLINEAR CASE , 2006, math/0601092.

[10]  Nawaf Bou-Rabee,et al.  Cayley modification for strongly stable path-integral and ring-polymer molecular dynamics. , 2019, The Journal of chemical physics.

[11]  Jianfeng Lu,et al.  Accelerated sampling by infinite swapping of path integral molecular dynamics with surface hopping. , 2017, The Journal of chemical physics.

[12]  Jianshu Cao,et al.  The formulation of quantum statistical mechanics based on the Feynman path centroid density. II. Dynamical properties , 1994 .

[13]  G. Roberts,et al.  MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster , 2012, 1202.0709.

[14]  Shiwei Lan,et al.  Adaptive dimension reduction to accelerate infinite-dimensional geometric Markov Chain Monte Carlo , 2018, J. Comput. Phys..

[15]  A. Stuart,et al.  Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions , 2011, 1112.1392.

[16]  W. Miller,et al.  A classical analog for electronic degrees of freedom in nonadiabatic collision processes , 1979 .

[17]  G. Martyna,et al.  Adiabatic path integral molecular dynamics methods. II. Algorithms , 1996 .

[18]  A. Eberle.  Error bounds for Metropolis–Hastings algorithms applied to perturbations of Gaussian measures in high dimensions , 2012, 1210.1180.

[19]  Thomas F. Miller,et al.  Path-integral isomorphic Hamiltonian for including nuclear quantum effects in non-adiabatic dynamics. , 2017, The Journal of chemical physics.

[20]  B. Leimkuhler,et al.  Robust and efficient configurational molecular sampling via Langevin dynamics. , 2013, The Journal of chemical physics.

[21]  J. Voss,et al.  Analysis of SPDEs arising in path sampling. Part I: The Gaussian case , 2005 .

[22]  Gregory A. Voth,et al.  A derivation of centroid molecular dynamics and other approximate time evolution methods for path integral centroid variables , 1999 .

[23]  B. Berne,et al.  Efficient molecular dynamics and hybrid Monte Carlo algorithms for path integrals , 1993 .

[24]  T. Matsubara.  A New Approach to Quantum-Statistical Mechanics , 1955 .

[25]  Jianfeng Lu,et al.  Path integral molecular dynamics with surface hopping for thermal equilibrium sampling of nonadiabatic systems. , 2017, The Journal of chemical physics.

[26]  Jian Liu,et al.  Path integral Liouville dynamics: Applications to infrared spectra of OH, water, ammonia, and methane. , 2016, The Journal of chemical physics.

[27]  Michael J. Willatt,et al.  Boltzmann-conserving classical dynamics in quantum time-correlation functions: "Matsubara dynamics". , 2015, The Journal of chemical physics.

[28]  Giovanni Bussi,et al.  Nuclear quantum effects in solids using a colored-noise thermostat. , 2009, Physical review letters.

[29]  D. Manolopoulos,et al.  An efficient ring polymer contraction scheme for imaginary time path integral simulations. , 2008, The Journal of chemical physics.

[30]  Glenn J. Martyna,et al.  Adiabatic path integral molecular dynamics methods. I. Theory , 1996 .

[31]  J. Zabczyk,et al.  Ergodicity for Infinite Dimensional Systems: Invariant measures for stochastic evolution equations , 1996 .

[32]  Peter G. Wolynes,et al.  Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids , 1981 .

[33]  Raphael Zimmer,et al.  Explicit contraction rates for a class of degenerate and infinite-dimensional diffusions , 2016, 1605.07863.

[34]  Jian Liu,et al.  Path integral molecular dynamics for exact quantum statistics of multi-electronic-state systems. , 2017, The Journal of chemical physics.

[35]  B. Simon.  Functional integration and quantum physics , 1979 .

[36]  Jesús María Sanz-Serna,et al.  Geometric integrators and the Hamiltonian Monte Carlo method , 2017, Acta Numerica.

[37]  A. Eberle,et al.  Couplings and quantitative contraction rates for Langevin dynamics , 2017, The Annals of Probability.

[38]  N. Pillai,et al.  A Function Space HMC Algorithm With Second Order Langevin Diffusion Limit , 2013, 1308.0543.

[39]  Andrew M. Stuart,et al.  Geometric MCMC for infinite-dimensional inverse problems , 2016, J. Comput. Phys..

[40]  Jian Liu,et al.  Path integral Liouville dynamics for thermal equilibrium systems. , 2014, The Journal of chemical physics.

[41]  Martin Hairer,et al.  The motion of a random string , 2016, 1605.02192.

[42]  Jian Liu,et al.  A simple and accurate algorithm for path integral molecular dynamics with the Langevin thermostat. , 2016, The Journal of chemical physics.

[43]  Michele Parrinello,et al.  Accelerating the convergence of path integral dynamics with a generalized Langevin equation. , 2011, The Journal of chemical physics.

[44]  Nawaf Bou-Rabee,et al.  Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019 .

[45]  G. Roberts,et al.  MCMC methods for diffusion bridges , 2008 .

[46]  J. Doll,et al.  A Monte Carlo method for quantum Boltzmann statistical mechanics using Fourier representations of path integrals , 1984 .

[47]  Roman Korol,et al.  Dimension-free path-integral molecular dynamics without preconditioning , 2020, The Journal of chemical physics.

[48]  Ian R. Craig,et al.  Quantum statistics and classical mechanics: real time correlation functions from ring polymer molecular dynamics. , 2004, The Journal of chemical physics.

[49]  J. M. Sanz-Serna,et al.  Hybrid Monte Carlo on Hilbert spaces , 2011 .

[50]  N Makri,et al.  Time-dependent quantum methods for large systems. , 1999, Annual review of physical chemistry.

[51]  Michael J. Willatt,et al.  Communication: Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics. , 2015, The Journal of chemical physics.