# Energy conservation in molecular dynamics simulations of classical systems

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### Resumé

Classical Newtonian dynamics is analytic and the energy of an isolated system is conserved. The energy of such a system, obtained by the discrete “Verlet” algorithm commonly used in molecular dynamics simulations, fluctuates but is conserved in the mean. This is explained by the existence of a “shadow Hamiltonian” [S. Toxvaerd, Phys. Rev. E 50, 2271 (1994)] , i.e., a Hamiltonian close to the original H with the property that the discrete positions of the Verlet algorithm for H lie on the analytic trajectories of . The shadow Hamiltonian can be obtained from H by an asymptotic expansion in the time step length. Here we use the first non-trivial term in this expansion to obtain an improved estimate of the discrete values of the energy. The investigation is performed for a representative system with Lennard-Jones pair interactions. The simulations show that inclusion of this term reduces the standard deviation of the energy fluctuations by a factor of 100 for typical values of the time step length. Simulations further show that the energy is conserved for at least one hundred million time steps provided the potential and its first four derivatives are continuous at the cutoff. Finally, we show analytically as well as numerically that energy conservation is not sensitive to round-off errors.
Originalsprog Engelsk Journal of Chemical Physics 136 22 224106-1 til 224106-8 8 0021-9606 https://doi.org/10.1063/1.4726728 Udgivet - 2012

### Citer dette

title = "Energy conservation in molecular dynamics simulations of classical systems",
abstract = "Classical Newtonian dynamics is analytic and the energy of an isolated system is conserved. The energy of such a system, obtained by the discrete “Verlet” algorithm commonly used in molecular dynamics simulations, fluctuates but is conserved in the mean. This is explained by the existence of a “shadow Hamiltonian” [S. Toxvaerd, Phys. Rev. E 50, 2271 (1994)] , i.e., a Hamiltonian close to the original H with the property that the discrete positions of the Verlet algorithm for H lie on the analytic trajectories of . The shadow Hamiltonian can be obtained from H by an asymptotic expansion in the time step length. Here we use the first non-trivial term in this expansion to obtain an improved estimate of the discrete values of the energy. The investigation is performed for a representative system with Lennard-Jones pair interactions. The simulations show that inclusion of this term reduces the standard deviation of the energy fluctuations by a factor of 100 for typical values of the time step length. Simulations further show that the energy is conserved for at least one hundred million time steps provided the potential and its first four derivatives are continuous at the cutoff. Finally, we show analytically as well as numerically that energy conservation is not sensitive to round-off errors.",
keywords = "classical mechanics, fluctuations, molecular dynamics method",
author = "S{\o}ren Toxv{\ae}rd and Ole Heilmann and Dyre, {J. C.}",
year = "2012",
doi = "10.1063/1.4726728",
language = "English",
volume = "136",
pages = "224106--1 til 224106--8",
journal = "Journal of Chemical Physics",
issn = "0021-9606",
publisher = "American Institute of Physics",
number = "22",

}

I: Journal of Chemical Physics, Bind 136, Nr. 22, 2012, s. 224106-1 til 224106-8.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Energy conservation in molecular dynamics simulations of classical systems

AU - Toxværd, Søren

AU - Heilmann, Ole

AU - Dyre, J. C.

PY - 2012

Y1 - 2012

N2 - Classical Newtonian dynamics is analytic and the energy of an isolated system is conserved. The energy of such a system, obtained by the discrete “Verlet” algorithm commonly used in molecular dynamics simulations, fluctuates but is conserved in the mean. This is explained by the existence of a “shadow Hamiltonian” [S. Toxvaerd, Phys. Rev. E 50, 2271 (1994)] , i.e., a Hamiltonian close to the original H with the property that the discrete positions of the Verlet algorithm for H lie on the analytic trajectories of . The shadow Hamiltonian can be obtained from H by an asymptotic expansion in the time step length. Here we use the first non-trivial term in this expansion to obtain an improved estimate of the discrete values of the energy. The investigation is performed for a representative system with Lennard-Jones pair interactions. The simulations show that inclusion of this term reduces the standard deviation of the energy fluctuations by a factor of 100 for typical values of the time step length. Simulations further show that the energy is conserved for at least one hundred million time steps provided the potential and its first four derivatives are continuous at the cutoff. Finally, we show analytically as well as numerically that energy conservation is not sensitive to round-off errors.

AB - Classical Newtonian dynamics is analytic and the energy of an isolated system is conserved. The energy of such a system, obtained by the discrete “Verlet” algorithm commonly used in molecular dynamics simulations, fluctuates but is conserved in the mean. This is explained by the existence of a “shadow Hamiltonian” [S. Toxvaerd, Phys. Rev. E 50, 2271 (1994)] , i.e., a Hamiltonian close to the original H with the property that the discrete positions of the Verlet algorithm for H lie on the analytic trajectories of . The shadow Hamiltonian can be obtained from H by an asymptotic expansion in the time step length. Here we use the first non-trivial term in this expansion to obtain an improved estimate of the discrete values of the energy. The investigation is performed for a representative system with Lennard-Jones pair interactions. The simulations show that inclusion of this term reduces the standard deviation of the energy fluctuations by a factor of 100 for typical values of the time step length. Simulations further show that the energy is conserved for at least one hundred million time steps provided the potential and its first four derivatives are continuous at the cutoff. Finally, we show analytically as well as numerically that energy conservation is not sensitive to round-off errors.

KW - classical mechanics

KW - fluctuations

KW - molecular dynamics method

U2 - 10.1063/1.4726728

DO - 10.1063/1.4726728

M3 - Journal article

VL - 136

SP - 224106-1 til 224106-8

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 22

ER -