Stationary state distribution and efficiency analysis of the Langevin equation via real or virtual dynamics

Li, Dezhang; Han, Xu; Chai, Yichen; Wang, Cong; Zhang, Zhijun; Chen, Zifei; Liu, Jian; Shao, Jiushu

doi:10.1063/1.4996204

Title: Stationary state distribution and efficiency analysis of the Langevin equation via real or virtual dynamics

Journal Article · Thu Nov 09 00:00:00 EST 2017 · Journal of Chemical Physics

DOI:https://doi.org/10.1063/1.4996204· OSTI ID:1497877

^[1];

^[1]; Chai, Yichen ^[1]; Wang, Cong ^[2];

^[1];

^[1]; Shao, Jiushu ^[2]

Peking University, Beijing (China). Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering
Beijing Normal University, Beijing (China). College of Chemistry and Center for Advanced Quantum Studies, Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education

Langevin dynamics has become a popular tool to simulate the Boltzmann equilibrium distribution. When the repartition of the Langevin equation involves the exact realization of the Ornstein-Uhlenbeck noise, in addition to the conventional density evolution, there exists another type of discrete evolution that may not correspond to a continuous, real dynamical counterpart. This virtual dynamics case is also able to produce the desired stationary distribution. Different types of repartition lead to different numerical schemes, of which the accuracy and efficiency are investigated through studying the harmonic oscillator potential, an analytical solvable model. By analyzing the asymptotic distribution and characteristic correlation time that are derived by either directly solving the discrete equations of motion or using the related phase space propagators, it is shown that the optimal friction coefficient resulting in the minimum characteristic correlation time depends on the time interval chosen in the numerical implementation. In conclusion, when the recommended “middle” scheme is employed, both analytical and numerical results demonstrate that, for good numerical performance in efficiency as well as accuracy, one may choose a friction coefficient in a wide range from around the optimal value to the high friction limit.

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Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: AC02-05CH11231

OSTI ID:: 1497877

Journal Information:: Journal of Chemical Physics, Vol. 147, Issue 18; ISSN 0021-9606

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 20 works

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References (39)

Algorithms for brownian dynamics van Gunsteren, W. F.; Berendsen, H. J. C. Molecular Physics, Vol. 45, Issue 3 https://doi.org/10.1080/00268978200100491	journal	February 1982
A fourth-order numerical integrator for stochastic Langevin equations Hershkovitz, Eli The Journal of Chemical Physics, Vol. 108, Issue 22 https://doi.org/10.1063/1.476380	journal	June 1998
A unified thermostat scheme for efficient configurational sampling for classical/quantum canonical ensembles via molecular dynamics Zhang, Zhijun; Liu, Xinzijian; Chen, Zifei The Journal of Chemical Physics, Vol. 147, Issue 3 https://doi.org/10.1063/1.4991621	journal	July 2017
Robust and efficient configurational molecular sampling via Langevin dynamics Leimkuhler, Benedict; Matthews, Charles The Journal of Chemical Physics, Vol. 138, Issue 17 https://doi.org/10.1063/1.4802990	journal	May 2013
Symplectic integrators from composite operator factorizations Chin, Siu A. Physics Letters A, Vol. 226, Issue 6 https://doi.org/10.1016/S0375-9601(97)00003-0	journal	March 1997
Brownian dynamics with hydrodynamic interactions Ermak, Donald L.; McCammon, J. A. The Journal of Chemical Physics, Vol. 69, Issue 4 https://doi.org/10.1063/1.436761	journal	August 1978
Path integrals for Fokker–Planck dynamics with singular diffusion: Accurate factorization for the time evolution operator Drozdov, Alexander N.; Talkner, Peter The Journal of Chemical Physics, Vol. 109, Issue 6 https://doi.org/10.1063/1.476721	journal	August 1998
Stochastic boundary conditions for molecular dynamics simulations of ST2 water Brünger, Axel; Brooks, Charles L.; Karplus, Martin Chemical Physics Letters, Vol. 105, Issue 5 https://doi.org/10.1016/0009-2614(84)80098-6	journal	March 1984
Rational Construction of Stochastic Numerical Methods for Molecular Sampling Leimkuhler, B.; Matthews, C. Applied Mathematics Research eXpress https://doi.org/10.1093/amrx/abs010	journal	June 2012
A Leap-frog Algorithm for Stochastic Dynamics Van Gunsteren, W. F.; Berendsen, H. J. C. Molecular Simulation, Vol. 1, Issue 3 https://doi.org/10.1080/08927028808080941	journal	March 1988
Quasilinear approximations for the propagator of the Fokker-Planck equation Drozdov, Alexander N. Physica A: Statistical Mechanics and its Applications, Vol. 196, Issue 2 https://doi.org/10.1016/0378-4371(93)90604-3	journal	June 1993
Efficient stochastic thermostatting of path integral molecular dynamics Ceriotti, Michele; Parrinello, Michele; Markland, Thomas E. The Journal of Chemical Physics, Vol. 133, Issue 12 https://doi.org/10.1063/1.3489925	journal	September 2010
Structure of positive decompositions of exponential operators Chin, Siu A. Physical Review E, Vol. 71, Issue 1 https://doi.org/10.1103/PhysRevE.71.016703	journal	January 2005
Algorithms for Brownian dynamics Ricci, Andrea; Ciccotti, Giovanni Molecular Physics, Vol. 101, Issue 12 https://doi.org/10.1080/0026897031000108113	journal	June 2003
Hybrid exponential product formulas for unbounded operators with possible applications to Monte Carlo simulations Suzuki, Masuo Physics Letters A, Vol. 201, Issue 5-6 https://doi.org/10.1016/0375-9601(95)00266-6	journal	June 1995
A simple and effective Verlet-type algorithm for simulating Langevin dynamics Grønbech-Jensen, Niels; Farago, Oded Molecular Physics, Vol. 111, Issue 8 https://doi.org/10.1080/00268976.2012.760055	journal	January 2013
Long-Run Accuracy of Variational Integrators in the Stochastic Context Bou-Rabee, Nawaf; Owhadi, Houman SIAM Journal on Numerical Analysis, Vol. 48, Issue 1 https://doi.org/10.1137/090758842	journal	January 2010
Construction of higher order symplectic integrators Yoshida, Haruo Physics Letters A, Vol. 150, Issue 5-7 https://doi.org/10.1016/0375-9601(90)90092-3	journal	November 1990
Path integral Liouville dynamics: Applications to infrared spectra of OH, water, ammonia, and methane Liu, Jian; Zhang, Zhijun The Journal of Chemical Physics, Vol. 144, Issue 3 https://doi.org/10.1063/1.4939953	journal	January 2016
Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics Suzuki, Masuo Journal of Mathematical Physics, Vol. 26, Issue 4 https://doi.org/10.1063/1.526596	journal	April 1985
The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics Leimkuhler, Benedict; Matthews, Charles; Stoltz, Gabriel IMA Journal of Numerical Analysis https://doi.org/10.1093/imanum/dru056	journal	January 2015
General theory of fractal path integrals with applications to many‐body theories and statistical physics Suzuki, Masuo Journal of Mathematical Physics, Vol. 32, Issue 2 https://doi.org/10.1063/1.529425	journal	February 1991
The determination of an accurate isotope dependent potential energy surface for water from extensive ab initio calculations and experimental data Partridge, Harry; Schwenke, David W. The Journal of Chemical Physics, Vol. 106, Issue 11 https://doi.org/10.1063/1.473987	journal	March 1997
Linearized semiclassical initial value time correlation functions using the thermal Gaussian approximation: Applications to condensed phase systems Liu, Jian; Miller, William H. The Journal of Chemical Physics, Vol. 127, Issue 11 https://doi.org/10.1063/1.2774990	journal	September 2007
Numerical integration of the Langevin equation: Monte Carlo simulation Ermak, Donald L.; Buckholz, Helen Journal of Computational Physics, Vol. 35, Issue 2 https://doi.org/10.1016/0021-9991(80)90084-4	journal	April 1980
Design of quasisymplectic propagators for Langevin dynamics Melchionna, Simone The Journal of Chemical Physics, Vol. 127, Issue 4 https://doi.org/10.1063/1.2753496	journal	July 2007
A simple and accurate algorithm for path integral molecular dynamics with the Langevin thermostat Liu, Jian; Li, Dezhang; Liu, Xinzijian The Journal of Chemical Physics, Vol. 145, Issue 2 https://doi.org/10.1063/1.4954990	journal	July 2016
Accurate sampling using Langevin dynamics Bussi, Giovanni; Parrinello, Michele Physical Review E, Vol. 75, Issue 5 https://doi.org/10.1103/PhysRevE.75.056707	journal	May 2007
Langevin stabilization of molecular dynamics Izaguirre, Jesús A.; Catarello, Daniel P.; Wozniak, Justin M. The Journal of Chemical Physics, Vol. 114, Issue 5 https://doi.org/10.1063/1.1332996	journal	February 2001
Monte Carlo Calculation of Quantum Systems. II. Higher Order Correction Takahashi, Minoru; Imada, Masatoshi Journal of the Physical Society of Japan, Vol. 53, Issue 11 https://doi.org/10.1143/JPSJ.53.3765	journal	November 1984
Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral Gillespie, Daniel T. Physical Review E, Vol. 54, Issue 2 https://doi.org/10.1103/PhysRevE.54.2084	journal	August 1996
High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schrödinger equation Skokos, Ch.; Gerlach, E.; Bodyfelt, J. D. Physics Letters A, Vol. 378, Issue 26-27 https://doi.org/10.1016/j.physleta.2014.04.050	journal	May 2014
Accurate sampling using Langevin dynamics Bussi, Giovanni; Parrinello, Michele arXiv https://doi.org/10.48550/arxiv.0803.4083	text	January 2008
Efficient stochastic thermostatting of path integral molecular dynamics Ceriotti, Michele; Parrinello, Michele; Markland, Thomas E. arXiv https://doi.org/10.48550/arxiv.1009.1045	text	January 2010
A simple and effective Verlet-type algorithm for simulating Langevin dynamics Grønbech-Jensen, Niels; Farago, Oded arXiv https://doi.org/10.48550/arxiv.1212.1244	text	January 2012
High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schroedinger equation Skokos, Ch.; Gerlach, E.; Bodyfelt, J. D. arXiv https://doi.org/10.48550/arxiv.1302.1788	text	January 2013
Robust and efficient configurational molecular sampling via Langevin Dynamics Leimkuhler, Benedict; Matthews, Charles arXiv https://doi.org/10.48550/arxiv.1304.3269	text	January 2013
The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics Leimkuhler, Benedict; Matthews, Charles; Stoltz, Gabriel arXiv https://doi.org/10.48550/arxiv.1308.5814	text	January 2013
A simple and accurate algorithm for path integral molecular dynamics with the Langevin thermostat Liu, Jian; Li, Dezhang; Liu, Xinzijian arXiv https://doi.org/10.48550/arxiv.1611.06331	text	January 2016

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Dimension-free path-integral molecular dynamics without preconditioning Korol, Roman; Rosa-Raíces, Jorge L.; Bou-Rabee, Nawaf The Journal of Chemical Physics, Vol. 152, Issue 10 https://doi.org/10.1063/1.5134810	journal	March 2020
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Complete set of stochastic Verlet-type thermostats for correct Langevin simulations Grønbech-Jensen, Niels Molecular Physics, Vol. 118, Issue 8 https://doi.org/10.1080/00268976.2019.1662506	journal	September 2019
Critical role of quantum dynamical effects in the Raman spectroscopy of liquid water Liu, Xinzijian; Liu, Jian Taylor & Francis https://doi.org/10.6084/m9.figshare.5897296	text	January 2018
Critical role of quantum dynamical effects in the Raman spectroscopy of liquid water Liu, Xinzijian; Liu, Jian arXiv https://doi.org/10.48550/arxiv.1712.10115	text	January 2017

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