Ensemble simulations with discrete classical dynamics

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.