Ensemble simulations with discrete classical dynamics.

Area of Science:

• Computational Physics
• Materials Science
• Chemical Physics

Background:

• Classical Molecular Dynamics (MD) simulations often employ the Verlet algorithm (VA) for numerical integration.
• The discrete nature of VA introduces energy drift, limiting the accuracy and time step size.
• Existing methods for discrete dynamics provide zero-order energy estimates (E0(h)).

Purpose of the Study:

• To develop a more accurate theoretical framework for discrete classical Molecular Dynamics (MD).
• To introduce a shadow Hamiltonian (H̃) that allows discrete particle positions to follow analytic trajectories.
• To derive a time-reversible VA algorithm for canonical dynamics (NVT̃(h)) and analyze energy/temperature relations across different ensembles.

Main Methods:

• Derivation of a shadow Hamiltonian (H̃) for discrete MD with the Verlet algorithm (VA).
• First-order estimation of the shadow Hamiltonian's energy (Ẽ(h)) and its relation to analytic dynamics energy (E).
• Development of a time-reversible VA algorithm for the canonical (NVT̃(h)) ensemble and comparison with (NVE0(h)) and (NVT0(h)) ensembles.

Main Results:

• The energy difference between discrete and analytic dynamics is proportional to h², with deviations of tenths of a percent for typical time increments (h).
• Relations between energies and temperatures for different ensembles ((NVẼ(h)) vs. (NVE), (NVT̃(h)) vs. (NVT)) are established.
• Accurate energy determination enables precise calculation of kinetic degrees of freedom (3N-3 for 3D systems).

Conclusions:

• The shadow Hamiltonian approach significantly improves the accuracy of discrete MD simulations.
• The derived relations allow for the use of larger time increments in MD, enhancing computational efficiency.
• Accurate kinetic energy calculations are crucial for simulating phenomena like nucleation in small systems.