Moving boundary truncated grid method: Application to activated barrier crossing with the Klein–Kramers and Boltzmann–BGK models

Abstract

We exploit the moving boundary truncated grid method for the Klein–Kramers and Boltzmann–BGK kinetic equations to approach the problem of thermally activated barrier crossing across non-parabolic barriers with reduced computational effort. The grid truncation algorithm dynamically deactivates the insignificant grid points while the boundary extrapolation procedure explores potentially important portions of phase space. An economized Eulerian framework is established to integrate the kinetic equations in the tailored phase space efficiently. The effects of coupling strength, kinetic model, and potential shape on the escape rate are assessed through direct numerical simulations. Besides, we adapt the Padé approximant approach for non-parabolic barriers by introducing a correction factor into the spatial diffusion asymptote to account for the anharmonicity. The modified Padé approximants are remarkably consistent with the numerical results obtained from the conventional full grid method in underdamped and overdamped regimes, whereas overestimating the rates in the turnover region, even exceeding the upper bound given by the transition-state theory. By contrast, the truncated grid method provides accurate rate estimates in excellent agreement with the full grid benchmarks globally, with negligible relative errors lower than 1.02% for the BGK model and below 0.56% for the Kramers model, while substantially reducing the computational cost. Overall, the truncated grid method has shown great promise as a high-performance scheme for the escape problem.