# Energy drift in molecular dynamics simulations

- 198 Downloads
- 6 Citations

## Abstract

In molecular dynamics, Hamiltonian systems of differential equations are numerically integrated using the Störmer–Verlet method. One feature of these simulations is that there is an unphysical drift in the energy of the system over long integration periods. We study this energy drift, by considering a representative system in which it can be easily observed and studied. We show that if the system is started in a random initial configuration, the error in energy of the numerically computed solution is well modeled as a continuous-time stochastic process: geometric Brownian motion. We discuss what in our model is likely to remain the same or to change if our approach is applied to more realistic molecular dynamics simulations.

## Key words

Hamiltonian systems symplectic numerical methods modified Hamiltonian shadow Hamiltonian backward error analysis molecular dynamics## Preview

Unable to display preview. Download preview PDF.

## References

- 1.M. P. Allen and D. J. Tildesley,
*Computer Simulation of Liquids*, Oxford University Press, Oxford, 1987.MATHGoogle Scholar - 2.G. Benettin and A. Giorgilli,
*On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms*, J. Stat. Phys., 74 (1993), pp. 1117–1142.CrossRefGoogle Scholar - 3.P. Billingsley,
*Convergence of Probability Measures*, John Wiley & Sons, Inc., New York, 1999.MATHGoogle Scholar - 4.G. Ciccotti, D. Frenkel, and I. R. McDonald, eds.,
*Simulation of Liquids and Solids: Molecular Dynamics and Monte Carlo Methods in Statistical Mechanics*, North-Holland, Amsterdam, 1987.Google Scholar - 5.D. Cottrell,
*Symplectic integration of simple collisions: A backward error analysis*, Master’s thesis, McGill University, 2004. (www.math.mcgill.ca/∼cottrell/thesis.pdf)Google Scholar - 6.E. Faou, E. Hairer, and T.-L. Pham,
*Energy conservation with non-symplectic methods: examples and counter-examples*, BIT, 44 (2004), pp. 699–709.MATHCrossRefGoogle Scholar - 7.E. Hairer, C. Lubich, and G. Wanner,
*Geometric Numerical Integration – Structure-Preserving Algorithms for Ordinary Differential Equations*, Springer, New York, 2002.MATHGoogle Scholar - 8.E. Hairer, C. Lubich, and G. Wanner,
*Geometric numerical integration illustrated by the Störmer–Verlet method*, Acta Numer., (2003), pp. 339–450.Google Scholar - 9.R. I. McLachlan and M. Perlmutter,
*Energy drift in reversible time integration*, J. Phys. A, Math. Gen., 37 (2004), pp. L593–L598.MATHCrossRefGoogle Scholar - 10.C. S. O’Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel,
*Jamming at zero temperature and zero applied stress: the epitome of disorder*, Phys. Rev. E, 68 (2003), pp. 1–19.CrossRefGoogle Scholar - 11.S. Reich,
*Backward error analysis for numerical integrators*, SIAM J. Numer. Anal., 36 (1999), pp. 1549–1570.MATHCrossRefGoogle Scholar - 12.E. Shaw, H. Sigurgeirsson, and A. M. Stuart,
*A Markov model for billiards*, preprint, Warwick Mathematics Institute, 2004.Google Scholar - 13.R. D. Skeel and D. J. Hardy,
*Practical construction of modified Hamiltonians*, SIAM J. Sci. Comput., 23 (2001), pp. 1172–1188.MATHCrossRefGoogle Scholar - 14.R. D. Skeel and D. J. Hardy,
*Monitoring energy drift with shadow Hamiltonians*, J. Comput. Phys., 206 (2005), pp. 432–452.MATHCrossRefGoogle Scholar - 15.L. Verlet,
*Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard–Jones molecules*, Phys. Rev., 159 (1967), pp. 98–103.Google Scholar