- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 3.0 License. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Using Nonequilibrium Fluctuation Theorems to Understand and Correct Errors in Equilibrium and Nonequilibrium Simulations of Discrete Langevin Dynamics

David A. Sivak1,*,†, John D. Chodera2,‡, and Gavin E. Crooks1

  • 1Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
  • 2California Institute of Quantitative Biosciences (QB3), University of California, Berkeley, California 94720, USA

  • *david.sivak@ucsf.edu
  • †Current address: Center for Systems and Synthetic Biology, University of California, San Francisco, California 94158, USA
  • ‡Current address: Computational Biology Program, Memorial Sloan-Kettering Cancer Center, New York, New York 10065, USA
PDF

Phys. Rev. X 3, 011007 – Published 29 January, 2013

DOI: https://doi.org/10.1103/PhysRevX.3.011007

Abstract

Common algorithms for computationally simulating Langevin dynamics must discretize the stochastic differential equations of motion. These resulting finite-time-step integrators necessarily have several practical issues in common: Microscopic reversibility is violated, the sampled stationary distribution differs from the desired equilibrium distribution, and the work accumulated in nonequilibrium simulations is not directly usable in estimators based on nonequilibrium work theorems. Here, we show that, even with a time-independent Hamiltonian, finite-time-step Langevin integrators can be thought of as a driven, nonequilibrium physical process. Once an appropriate worklike quantity is defined—here called the shadow work—recently developed nonequilibrium fluctuation theorems can be used to measure or correct for the errors introduced by the use of finite time steps. In particular, we demonstrate that amending estimators based on nonequilibrium work theorems to include this shadow work removes the time-step-dependent error from estimates of free energies. We also quantify, for the first time, the magnitude of deviations between the sampled stationary distribution and the desired equilibrium distribution for equilibrium Langevin simulations of solvated systems of varying sizes. While these deviations can be large, they can be eliminated altogether by Metropolization or greatly diminished by small reductions in the time step. Through this connection with driven processes, further developments in nonequilibrium fluctuation theorems can provide additional analytical tools for dealing with errors in finite-time-step integrators.

View figure in article

Popular Summary

Enabled partly by the breathtaking developments in computer technology, simulations on modern computers of the dynamics of small and large atomic or molecular systems are ubiquitous and indispensable in physics, chemistry, and biology, for the testing of new theories or the modeling of experiments that can’t be done in a lab. But, such simulations suffer inevitably some inaccuracies or artifacts. One source of artifacts comes from approximating the continuous passage of time with a digitized (or discretized) series of time slices. To develop a physical understanding of this source of artifacts and to systematically control or correct such artifacts in simulations is an ongoing quest of the computational physics community. In this paper, we achieve this goal in the context of simulations of Langevin dynamics—a workhorse theory for describing molecular dynamics. We do this through a new cross-topic perspective that brings together recent fundamental developments in nonequilibrium statistical mechanics and computer simulations.

Langevin dynamics, first developed about a century ago by French physicist Paul Langevin, is a simple theory for describing motions—at molecular scales or beyond—of effective particles that are not necessarily the constituent atoms or molecules themselves. It lumps faster and shorter-scale motions than those described into an effective friction and a series of random kicks on the effective particles. We used this theory to model the behavior of simple model systems, and confirmed what had already been shown mathematically by others: that computer simulations based on the time slices of the dynamics deviated significantly from what the exact theory predicted.

Our new insight is a physical interpretation of this otherwise-cryptic result: that the use of time slices has an effect analogous to doing additional work on the system—which we call “shadow work” because the system behaves as if there is an additional nonequilibrium driving force, one that is not explicitly modeled. This insight, confirmed by simulations of a toy model of one effective particle, leads to further understanding and methodological progress. Simulations of a more complex system consisting of hundreds or thousands of water molecules permitted a measurement of how big the error is. With an appropriate algorithm, we are able to isolate this shadow work from the physically meaningful work, quantify it, and, in turn, systematically correct for it, using recently developed theorems from the field of nonequilibrium statistical mechanics.

We are extremely optimistic about future developments in this new direction. We anticipate that other insights from nonequilibrium statistical mechanics can be exploited to the gain of computer simulations, and we also believe that other methods that simulate molecular dynamics with related but more complicated schemes than the straightforward Langevin dynamics may benefit from this perspective, too.

Article Text

References (60)

  1. Paul Langevin, Sur la theórie du mouvement Brownien, C. R. Acad. Sci. (Paris) 146, 530 (1908) [English translation: On the Theory of Brownian Motion, Am. J. Phys. 65, 1079 (1997)].
  2. Daan Frenkel and Berend Smit, Understanding Molecular Simulation (Academic Press, San Diego, 2002), 2nd ed.
  3. Manuel Athènes, A Path-Sampling Scheme for Computing Thermodynamic Properties of a Many-Body System in a Generalized Ensemble, Eur. Phys. J. B 38, 651 (2004).
  4. Gilles Adjanor and Manuel Athènes, Gibbs Free-Energy Estimates from Direct Path-Sampling Computations, J. Chem. Phys. 123, 234104 (2005).
  5. Gilles Adjanor, Manuel Athènes, and Florent Calvo, Free Energy Landscape from Path-Sampling: Application to the Structural Transition in LJ38, Eur. Phys. J. B 53, 47 (2006).
  6. Wolfgang Lechner, Harald Oberhofer, Christoph Dellago, and Phillip L. Geissler, Equilibrium Free Energies from Fast-Switching Trajectories with Large Time Steps, J. Chem. Phys. 124, 044113 (2006).
  7. William C. Swope, Hans C. Andersen, Peter H. Berens, and Kent R. Wilson, A Computer Simulation Method for the Calculation of Equilibrium Constants for the Formation of Physical Clusters of Molecules: Application to Small Water Clusters, J. Chem. Phys. 76, 637 (1982).
  8. Mark E. Tuckerman, Bruce J. Berne, and Glenn J. Martyna, Reversible Multiple Time Scale Molecular Dynamics, J. Chem. Phys. 97, 1990 (1992).
  9. J. M. Sanz-Serna, Symplectic Integrators for Hamiltonian Problems: An Overview, Acta Numer. 1, 243 (1992) [published online 2008].
  10. H. Yoshida, Construction of Higher Order Symplectic Integrators, Phys. Lett. A 150, 262 (1990).
  11. Giovanni Bussi and Michele Parrinello, Accurate Sampling Using Langevin Dynamics, Phys. Rev. E 75, 056707 (2007).
  12. Tony Lelièvre, Mathias Rousset, and Gabriel Stoltz, Free Energy Computations: A Mathematical Perspective (Imperial College Press, London, 2010).
  13. Richard C. Tolman, The Principles of Statistical Mechanics (Oxford University Press, London, 1938).
  14. Giovanni Bussi, Tatyana Zykova-Timan, and Michele Parrinello, Isothermal-Isobaric Molecular Dynamics Using Stochastic Velocity Rescaling, J. Chem. Phys. 130, 074101 (2009).
  15. Nawaf Bou-Rabee and Houman Owhadi, Long-Run Behavior of Variational Integrators in the Stochastic Context, SIAM J. Numer. Anal. 48, 278 (2010).
  16. Tony Lelièvre, Mathias Rousset, and Gabriel Stoltz, Langevin Dynamics with Constraints and Computation of Free Energy Differences, Math. Comput. 81, 2071 (2012).
  17. Carlos Bustamante, Jan Liphardt, and Felix Ritort, The Nonequilibrium Thermodynamics of Small Systems, Phys. Today 58, 43 (2005).
  18. B. Cleuren, C. Van den Broeck, and R. Kawai, Fluctuation and Dissipation, C.R. Physique 8, 567 (2007).
  19. Christopher Jarzynski, Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale, Annu. Rev. Condens. Matter Phys. 2, 329 (2011).
  20. Richard E. Spinney and Ian J. Ford, in Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond, edited by R. Klages, W. Just, and C. Jarzynski (Wiley-VCH, Weinheim, 2013), Chap. 1, p. 3 [to be published; ISBN: [Amazon][WorldCat]].
  21. Gavin E. Crooks, Nonequilibrium Measurements of Free Energy Differences for Microscopically Reversible Markovian Systems, J. Stat. Phys. 90, 1481 (1998).
  22. G. E. Crooks, Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences, Phys. Rev. E 60, 2721 (1999).
  23. G. E. Crooks, Excursions in Statistical Dynamics, Ph.D. thesis, University of California, Berkeley, 1999.
  24. Gavin E. Crooks, On Thermodynamic and Microscopic Reversibility, J. Stat. Mech. (2011) P07008.
  25. Christopher Jarzynski, Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett. 78, 2690 (1997).
  26. Denis J. Evans and Debra J. Searles, Equilibrium Microstates Which Generate Second Law Violating Steady-States, Phys. Rev. E 50, 1645 (1994).
  27. Gerhard Hummer and Attila Szabo, Free Energy Reconstruction from Nonequilibrium Single-Molecule Pulling Experiments, Proc. Natl. Acad. Sci. U.S.A. 98, 3658 (2001).
  28. Pierre Gaspard, Time-Reversed Dynamical Entropy and Irreversibility in Markovian Random Processes, J. Stat. Phys. 117, 599 (2004).
  29. Christopher Jarzynski, Rare Events and the Convergence of Exponentially Averaged Work Values, Phys. Rev. E 73, 046105 (2006).
  30. R. Kawai, J. M. R. Parrondo, and C. Van den Broeck, Dissipation: The Phase-Space Perspective, Phys. Rev. Lett. 98, 080602 (2007).
  31. Edward H. Feng and Gavin E. Crooks, Length of Time’s Arrow, Phys. Rev. Lett. 101, 090602 (2008).
  32. David A. Sivak and Gavin E. Crooks, Near-Equilibrium Measurements of Nonequilibrium Free Energy, Phys. Rev. Lett. 108, 150601 (2012).
  33. Robert Shaw, The Dripping Faucet as a Model Chaotic System (Aerial Press, Santa Cruz, 1984).
  34. Bernard Gaveau and L. S. Schulman, A General Framework for Non-Equilibrium Phenomena: The Master Equation and Its Formal Consequences, Phys. Lett. A 229, 347 (1997).
  35. Alan M. Horowitz, A Generalized Guided Monte Carlo Algorithm, Phys. Lett. B 268, 247 (1991).
  36. A. Beskos, N. S. Pillai, G. O. Roberts, J. M. Sanz-Serna, and A. M. Stuart, Optimal Tuning of the Hybrid Monte-Carlo Algorithm, arXiv:1001.4460.
  37. Reinaldo García-García, Daniel Domínguez, Vivien Lecomte, and Alejandro B. Kolton, Unifying Approach for Fluctuation Theorems from Joint Probability Distributions, Phys. Rev. E 82, 030104 (2010).
  38. Reinaldo García-García, Vivien Lecomte, Alejandro B. Kolton, and Daniel Domínguez, Joint Probability Distributions and Fluctuation Theorems, J. Stat. Mech. (2012) P02009.
  39. Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller, Equation of State Calculations by Fast Computing Machines, J. Chem. Phys. 21, 1087 (1953).
  40. Manuel Athènes, Computation of a Chemical Potential Using a Residence Weight Algorithm, Phys. Rev. E 66, 046705 (2002).
  41. Jerome P. Nilmeier, Gavin E. Crooks, David D. L. Minh, and John D. Chodera, Nonequilibrium Candidate Monte Carlo Is an Efficient Tool for Equilibrium Simulation, Proc. Natl. Acad. Sci. U.S.A. 108, E1009 (2011).
  42. Sean X. Sun, Equilibrium Free Energies from Path Sampling of Nonequilibrium Trajectories, J. Chem. Phys. 118, 5769 (2003).
  43. Erdinç Atılgan and Sean X. Sun, Equilibrium Free Energy Estimates Based on Nonequilibrium Work Relations and Extended Dynamics, J. Chem. Phys. 121, 10392 (2004).
  44. David L. D. Minh and John D. Chodera, Optimal Estimators and Asymptotic Variances for Nonequilibrium Path-Ensemble Averages, J. Chem. Phys. 131, 134110 (2009).
  45. David L. D. Minh and John D. Chodera, Estimating Equilibrium Ensemble Averages Using Multiple Time Slices from Driven Nonequilibrium Processes: Theory and Application to Free Energies, Moments, and Thermodynamic Length in Single-Molecule Pulling Experiments, J. Chem. Phys. 134, 024111 (2011).
  46. Takahiro Sagawa and Masahito Ueda, Generalized Jarzynski Equality under Nonequilibrium Feedback Control, Phys. Rev. Lett. 104, 090602 (2010).
  47. Gary S. Ayton, Denis J. Evans, and Debra J. Searles, A Local Fluctuation Theorem, J. Chem. Phys. 115, 2033 (2001).
  48. David A. Sivak and Gavin E. Crooks, Thermodynamic Metrics and Optimal Paths, Phys. Rev. Lett. 108, 190602 (2012).
  49. David A. Sivak, John D. Chodera, and Gavin E. Crooks, Time Step Rescaling Recovers Continuous-Time Dynamical Properties for Discrete-Time Langevin Integration of Nonequilibrium Systems, arXiv:1301.3800.
  50. Nawaf Bou-Rabee and Eric Vanden-Eijnden, Pathwise Accuracy and Ergodicity of Metropolized Integrators for SDEs, Commun. Pure Appl. Math. 63, 655 (2009).
  51. Mark S. Friedrichs, Peter Eastman, Vishal Vaidyanathan, Mike Houston, Scott LeGrand, Adam L. Beberg, Daniel L. Ensign, Christopher M. Bruns, and Vijay S. Pande, Accelerating Molecular Dynamic Simulation on Graphics Processing Units, J. Comput. Chem. 30, 864 (2009).
  52. Peter K. Eastman, Mark S. Friedrichs, John D. Chodera, Randall J. Radmer, Christopher M. Bruns, Joy P. Ku, Kyle A. Beauchamp, Thomas J. Lane, Lee-Ping Wang, Diwakar Shukla, Tony Tye, Michael Houston, Timo Stich, Christoph Klein, Michael R. Shirts, and Vijay S. Pande, OpenMM 4: A Reusable, Extensible Hardware Independent Library for High Performance Molecular Simulation, J. Chem. Theory Comput. 12 (2012) Articles ASAP.
  53. William L. Jorgensen, Jayaraman Chandrasekhar, Jeffrey D. Madura, Roger W. Impey, and Michael L. Klein, Comparison of Simple Potential Functions for Simulating Liquid Water, J. Chem. Phys. 79, 926 (1983).
  54. Shuichi Miyamoto and Peter A. Kollman, SETTLE: An Analytical Version of the SHAKE and RATTLE Algorithm for Rigid Water Models, J. Comput. Chem. 13, 952 (1992).
  55. Hans C. Andersen, RATTLE—A Velocity Version of the SHAKE Algorithm for Molecular-Dynamics Calculations, J. Comput. Phys. 52, 24 (1983).
  56. Michael R. Shirts, David L. Mobley, John D. Chodera, and Vijay S. Pande, Accurate and Efficient Corrections for Missing Dispersion Interactions in Molecular Simulations, J. Phys. Chem. B 111, 13052 (2007).
  57. Ilario G. Tironi, René Sperb, Paul E. Smith, and Wilfred F. van Gunsteren, A Generalized Reaction Field Method for Molecular Dynamics Simulations, J. Chem. Phys. 102, 5451 (1995).
  58. Kim-Hung Chow and David M. Ferguson, Isothermal-Isobaric Molecular Dynamics Simulations with Monte Carlo Volume Sampling, Comput. Phys. Commun. 91, 283 (1995).
  59. Johan Åqvist, Petra Wennerström, Martin Nervall, Sinisa Bjelic, and Bjørn O. Brandsdal, Molecular Dynamics Simulations of Water and Biomolecules with a Monte Carlo Constant Pressure Algorithm, Chem. Phys. Lett. 384, 288 (2004).
  60. John D. Chodera, William C. Swope, Jed W. Pitera, Chaok Seok, and Ken A. Dill, Use of the Weighted Histogram Analysis Method for the Analysis of Simulated and Parallel Tempering Simulations, J. Chem. Theory Comput. 3, 26 (2007).

Outline

Information

Phys. Rev. X 3, 011007– Published 29 January, 2013
  • Received 12 December 2011
Article part of CHORUS
Creative Commons Logo

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation