Shadow Hamiltonian in molecular dynamics simulations: Against a possible suggested misuse of its physical meaning

Symplectic integrators are largely employed in MD due to their long-term stability and near conservation of invariants. Their numerical robustness can be rationalized by the existence of a shadow HamiltonianHsh, exactly conserved by symplectic integrators and expressible as an asymptotic expansion with respect to the integration time steps δt around the physical Hamiltonian H. In this work, we suggest not attaching any physical meaning to Hsh, based on theoretical analysis and MD simulations of a standard LJ system using the velocity-Verlet algorithm. We do not intend to optimize symplectic algorithms. Instead, we claim that the existence of Hsh cannot provide better observables to estimate statistical properties. For that reason, we will also examine the behavior of trajectories integrated with large δt that are always exact integrations of their corresponding Hsh(δt). To these trajectories, of course, we do not attach a physical meaning. Note that the stability of the energy cannot be enough to guarantee accurate estimates of statistical properties. Although symplectic integrators yield, within numerical accuracy, exact trajectories of Hsh, thermodynamic quantities derived from Hsh deviate from the physical ones when δt is too large. Indeed, the conventional kinetic temperature of H is wrongly estimated for large δt, while "shadow" temperature becomes unphysical for the same large δt, despite its exact meaning for Hsh. We conclude that Hsh lacks physical relevance for large δt, and the only remedy to perform simulations by standard integrators, such as the velocity-Verlet, is to use a short enough δt, in which case the estimated physical properties coincide.