Ensemble simulations with discrete classical dynamics.

For discrete classical Molecular Dynamics (MD) obtained by the "Verlet" algorithm (VA) with the time increment h there exists (for sufficiently small h) a shadow Hamiltonian H̃ with energy Ẽ(h), for which the discrete particle positions lie on the analytic trajectories for H̃. The first order estimate of Ẽ(h) is employed to determine the relation with the corresponding energy, E, for the analytic dynamics with h = 0 and the zero-order estimate E0(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 (NVT̃(h)) ensemble and determine the relations between the energies and temperatures for the different ensembles, including the (NVE0(h)) and (NVT0(h)) ensembles. The differences in the energies and temperatures are proportional with h(2) and they are of the order of a few tenths of a percent for a traditional value of h. The relations between (NVẼ(h)) and (NVE), and (NVT̃(h)) and (NVT) are easily determined for a given density and temperature, and allow 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.