Extended phase-space symplectic integration for electron dynamics

TL;DR

This work develops and tests extended phase-space symplectic integration, a framework that enables high-order split-operator propagation for Hamiltonians that couple conjugate variables or possess beatified Poisson brackets. By duplicating phase-space coordinates and introducing an autonomized time variable, the method preserves symplectic structure while allowing analytical flows for the split components, with a computable on-the-fly accuracy metric given by the distance between copies. The authors apply the approach to two distinct systems: a classical ExB drift model with 1.5 DOF and KS TDDFT with infinite DOF, demonstrating similar stability and accuracy patterns and providing practical guidelines for choosing the restrain coefficient omega. They show that split-operator schemes with careful potential splitting and high-order accuracy achieve robust performance, expanding the reach of symplectic integration to both finite and infinite dimensional Hamiltonian dynamics and enabling efficient simulations of complex plasma and electronic dynamics.

Abstract

We investigate the use of extended phase-space symplectic integration [M. Tao, Phys. Rev. E 94, 043303 (2016)] for simulating two different classes of electron dynamics. The first one, with one and a half degrees of freedom, comes from plasma physics and describes the classical dynamics of a charged particle in a strong, constant, and uniform magnetic field perturbed by a turbulent electrostatic potential. The second one, with an infinite number of degrees of freedom, comes from physical chemistry and corresponds to Kohn-Sham time-dependent density-functional theory. For both we lay out the extension procedure and stability condition for numerical integration of the dynamics using high-order symplectic split-operator schemes. We also identify a computationally inexpensive metric that can be used for on-the-fly estimation of the accuracy of simulations. Our work paves the way for broad application of symplectic split-operator integration of classical and quantum Hamiltonian systems with finite and infinite number of degrees of freedom.

Figures (6)

• Figure 1: Comparison of the error in energy given by Eq. \ref{['eq:energy_conservation']} (blue) and averaged distance between copies (black) of extended phase-space symplectic integrations as functions of (a) the time step and (b) run time (in seconds). For all calculations, we use the optimized 4$^\text{th}$ order Blanes and Moan scheme and a restrain coefficient $\omega=10$. The ${\bf E}\times {\bf B}$ model parameters are $A=0.6$ and $M=25$. We use 500 trajectories integrated over 500 periods of the electrostatic field. All quantities are dimensionless.
• Figure 2: Comparison of the error in energy given by Eq. \ref{['eq:energy_conservation']} (upper panel) and distance between copies (black) of extended phase-space symplectic integrations as functions of $\omega$ for 3 different time steps: $dt=10^{-1}$ (thin line), $dt=5~10^{-2}$ (regular), and $dt=10^{-2}$ (bold). We use the same split-operator scheme, model parameters, and trajectory set as in Fig. \ref{['fig:ExB1']}.
• Figure 3: Critical restrain coefficient ${\rm max}_t\ \omega_c(x,y,t)$ given by Eq. \ref{['eq:ExB_wc']}, calculated from the local-stability analysis of section \ref{['sec:Methods:restrain_influence']}, as a function of $x$ and $y$. We use the same model parameters as in Figs. \ref{['fig:ExB1']} and \ref{['fig:ExB2']}.
• Figure 4: Comparison of the (a) accuracy and (b) efficacy of extended phase-space symplectic integrations using different types and orders of terms in the split operator -- see legend and main text. The HHR and TVR curves are nearly on top of each other. For all calculations, we use the optimized 4$^\text{th}$ order Blanes and Moan scheme and a restrain coefficient $\omega=10$. For reference, the dashed black line in panel (a) mark a perfect 4$^\text{th}$ order scaling.
• Figure 5: Comparison of the (a) accuracy and (b) distance between the extended phase-space Kohn-Sham orbitals \ref{['eq:TDDFT_extended_phase_space_distance']} while varying the restrain coefficient $\omega$ for different propagation time steps -- see labels on the curves. For all calculations, we use the optimized 4$^\text{th}$ order Blanes and Moan scheme and TRV operator split.