# Ensemble simulations with discrete classical dynamics

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

### Resumé

For discrete classical Molecular dynamics (MD) obtained by the "Verlet" algorithm (VA) with the time increment $h$ there exist a shadow Hamiltonian $\tilde{H}$ with energy $\tilde{E}(h)$, for which the discrete particle positions lie on the analytic trajectories for $\tilde{H}$. $\tilde{E}(h)$ is employed to determine the relation with the corresponding energy, $E$ for the analytic dynamics with $h=0$ and the zero-order estimate $E_0(h)$ of the energy for discrete dynamics, appearing in the literature for MD with VA. We derive a corresponding time reversible VA algorithm for canonical dynamics for the $(NV\tilde{T}(h))$ ensemble
and determine the relations between the energies and temperatures for the different ensembles, including the $(NVE_0(h))$ and $(NVT_0(h))$ ensembles.
The differences in the energies and temperatures are proportional with $h^2$ and they are of the order a few tenths of a percent for a traditional value of $h$.
The relations between $(NV\tilde{E}(h))$ and $(NVE)$, and $(NV\tilde{T}(h))$ and $(NVT)$ are easily determined for a given density and temperature, and allows for using larger time increments in MD. The accurate determinations of the energies are used to determine the kinetic degrees of freedom in a system of $N$ particles. It is $3N-3$ for a three dimensional system. The knowledge of the degrees of freedom is necessary when simulating small system, e.g. at nucleation.
Originalsprog Engelsk 139 Journal of Chemical Physics 139 22 224106-1 8 0021-9606 https://doi.org/10.1063/1.4836615 Udgivet - 14 dec. 2013

### Emneord

• dynamik ved konstant temperatur

### Citer dette

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title = "Ensemble simulations with discrete classical dynamics",
abstract = "For discrete classical Molecular dynamics (MD) obtained by the {"}Verlet{"} algorithm (VA) with the time increment $h$ there exist a shadow Hamiltonian $\tilde{H}$ with energy $\tilde{E}(h)$, for which the discrete particle positions lie on the analytic trajectories for $\tilde{H}$. $\tilde{E}(h)$ is employed to determine the relation with the corresponding energy, $E$ for the analytic dynamics with $h=0$ and the zero-order estimate $E_0(h)$ of the energy for discrete dynamics, appearing in the literature for MD with VA. We derive a corresponding time reversible VA algorithm for canonical dynamics for the $(NV\tilde{T}(h))$ ensembleand determine the relations between the energies and temperatures for the different ensembles, including the $(NVE_0(h))$ and $(NVT_0(h))$ ensembles.The differences in the energies and temperatures are proportional with $h^2$ and they are of the order a few tenths of a percent for a traditional value of $h$.The relations between $(NV\tilde{E}(h))$ and $(NVE)$, and $(NV\tilde{T}(h))$ and $(NVT)$ are easily determined for a given density and temperature, and allows for using larger time increments in MD. The accurate determinations of the energies are used to determine the kinetic degrees of freedom in a system of $N$ particles. It is $3N-3$ for a three dimensional system. The knowledge of the degrees of freedom is necessary when simulating small system, e.g. at nucleation.",
keywords = "dynamik ved konstant temperatur",
author = "S{\o}ren Toxv{\ae}rd",
year = "2013",
month = "12",
day = "14",
doi = "10.1063/1.4836615",
language = "English",
volume = "139",
pages = "224106--1",
journal = "Journal of Chemical Physics",
issn = "0021-9606",
publisher = "American Institute of Physics",
number = "22",

}

I: Journal of Chemical Physics, Bind 139, Nr. 22, 139, 14.12.2013, s. 224106-1.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Ensemble simulations with discrete classical dynamics

AU - Toxværd, Søren

PY - 2013/12/14

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N2 - For discrete classical Molecular dynamics (MD) obtained by the "Verlet" algorithm (VA) with the time increment $h$ there exist a shadow Hamiltonian $\tilde{H}$ with energy $\tilde{E}(h)$, for which the discrete particle positions lie on the analytic trajectories for $\tilde{H}$. $\tilde{E}(h)$ is employed to determine the relation with the corresponding energy, $E$ for the analytic dynamics with $h=0$ and the zero-order estimate $E_0(h)$ of the energy for discrete dynamics, appearing in the literature for MD with VA. We derive a corresponding time reversible VA algorithm for canonical dynamics for the $(NV\tilde{T}(h))$ ensembleand determine the relations between the energies and temperatures for the different ensembles, including the $(NVE_0(h))$ and $(NVT_0(h))$ ensembles.The differences in the energies and temperatures are proportional with $h^2$ and they are of the order a few tenths of a percent for a traditional value of $h$.The relations between $(NV\tilde{E}(h))$ and $(NVE)$, and $(NV\tilde{T}(h))$ and $(NVT)$ are easily determined for a given density and temperature, and allows for using larger time increments in MD. The accurate determinations of the energies are used to determine the kinetic degrees of freedom in a system of $N$ particles. It is $3N-3$ for a three dimensional system. The knowledge of the degrees of freedom is necessary when simulating small system, e.g. at nucleation.

AB - For discrete classical Molecular dynamics (MD) obtained by the "Verlet" algorithm (VA) with the time increment $h$ there exist a shadow Hamiltonian $\tilde{H}$ with energy $\tilde{E}(h)$, for which the discrete particle positions lie on the analytic trajectories for $\tilde{H}$. $\tilde{E}(h)$ is employed to determine the relation with the corresponding energy, $E$ for the analytic dynamics with $h=0$ and the zero-order estimate $E_0(h)$ of the energy for discrete dynamics, appearing in the literature for MD with VA. We derive a corresponding time reversible VA algorithm for canonical dynamics for the $(NV\tilde{T}(h))$ ensembleand determine the relations between the energies and temperatures for the different ensembles, including the $(NVE_0(h))$ and $(NVT_0(h))$ ensembles.The differences in the energies and temperatures are proportional with $h^2$ and they are of the order a few tenths of a percent for a traditional value of $h$.The relations between $(NV\tilde{E}(h))$ and $(NVE)$, and $(NV\tilde{T}(h))$ and $(NVT)$ are easily determined for a given density and temperature, and allows for using larger time increments in MD. The accurate determinations of the energies are used to determine the kinetic degrees of freedom in a system of $N$ particles. It is $3N-3$ for a three dimensional system. The knowledge of the degrees of freedom is necessary when simulating small system, e.g. at nucleation.

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