Refinement of thermostated molecular dynamics using backward error analysis

Abstract

Kinetic energy equipartition is a premise for many deterministic and stochastic molecular dynamics methods that aim at sampling a canonical ensemble. While this is expected for real systems, discretization errors introduced by the numerical integration may lead to deviations from equipartition. Fortunately, backward error analysis allows us to obtain a higher-order estimate of the quantity that is actually subject to equipartition. This is related to a shadow Hamiltonian, which converges to the specified Hamiltonian only when the time-step size approaches zero. This paper deals with discretization effects in a straightforward way. With a small computational overhead, we obtain refined versions of the kinetic and potential energies, whose sum is a suitable estimator of the shadow Hamiltonian. Then, we tune the thermostatting procedure by employing the refined kinetic energy instead of the conventional one. This procedure is shown to reproduce a canonical ensemble compatible with the refined system, as opposed to the original one, but canonical averages regarding the latter can be easily recovered by reweighting. Water, modeled as a rigid body, is an excellent test case for our proposal because its numerical stability extends up to time steps large enough to yield pronounced discretization errors in Verlet-type integrators. By applying our new approach, we were able to mitigate discretization effects in equilibrium properties of liquid water for time-step sizes up to 5 fs.

Molecular Dynamics with Very Large Time Steps for the Calculation of Solvation Free Energies

“…Once again, increasing the outer step size causes small variations in the free energies obtained. The curves suggest systematic deviations, as expected due to the existence of discretization effects, , but the magnitude of the uncertainties and the presence of some possible outliers make it difficult to draw an unequivocal conclusion.…”

“…Both the scaling and softcore approaches result in profiles that converge, within statistical sampling error, to the same final free energy. Due to time step discretization effects and the use of different outer time steps, a systematic component is expected to exist in these deviations. , It is important to determine whether such a component can be overlooked in practice (especially in the case of more complex molecules, as will be discussed in Section ). For this, we can analyze the computed solvation free energy as a function of the time step size.…”

MD Simulation of Water Using a Rigid Body Description Requires a Small Time Step to Ensure Equipartition

“…As noted in the Introduction, after we had posted a preprint of the present study, we became aware of careful investigations by Silveira and Abreu , and Davidchack that have a bearing on this work. Both groups had already noted the kind of discrepancy we find in Figure .…”

“…However, as their work makes clear, removing a drift in total energy is not enough to guarantee that equipartition is satisfied. In a subsequent study, Silveira and Abreu have also presented a refined version of kinetic and potential energies that can serve as a better estimator of the shadow Hamiltonian (also see ref in this regard). Our work here confirms the central observations of the breakdown of equipartition noted in these earlier studies.…”

“…This examination leads to the finding that for δt ≥ 1 fs, equipartition between translation and rotation is not satisfied, with the problem made worse as δt increases. (We note here that after we had posted a preprint of our work, we became aware of earlier works by Silveira and Abreu 12,13 and Davidchack. 14 These authors had found the same discrepancy between translational and rotational temperatures as here; we shall return to these important studies below.)…”