Dissipative Matsubara Dynamics

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The two most widely used path-integral chemical dynamics methods are Ring-polymer molecular dynamics (RPMD) and Centroid molecular dynamics (CMD). Both of these were originally proposed as heuristic approximations, which were justified by analytic limits and empirical evidence.In 2015, the Matsubara-dynamics theory was proposed as a controlled minimal approximation combining quantum statistics with classical dynamics, while conserving the quantum Boltzmann distribution. It was then shown that both RPMD and CMD are approximations to Matsubara dynamics, thus providing a firm theoretical ground for these methods. Unfortunately, a naive implementation of Matsubara dynamics is too computationally expensive to be a useful method because of a severe sign problem, which limited the number of Matsubara modes in a calculation to around 10 or less. Therefore, Matsubara dynamics has never been directly compared to exact quantum results with the exception of the harmonic oscillator and a linear correlation function cropped with a window function.

In this work, we present results for up to 200 Matsubara modes and fully converged non-linear correlation functions. This was achieved by developing dissipative Matsubara dynamics — a way of implicitly including a bath of harmonic oscillators in a Matsubara dynamics simulation.We first show that the fact that the bath is harmonic allows its inclusion in the simulation without exacerbating the sign problem. Then we proceed to show that the presence of the bath even allows simulations of analytically continued Matsubara dynamics, which does not suffer from the sign problem but was previously impossible due to the presence of unstable trajectories. These simulations allow the inclusion of almost an order of magnitude more Matsubara modes than previously feasible. To further improve the stability, we introduce the “real-noise” approximation, which allows the simulation of up to ≈ 200 Matsubara modes. However, even this number is insufficient to converge non-linear operators. Therefore, we developed a harmonic correction for the tail of the Matsubara distribution, with which we were able to obtain converged non-linear correlation functions, which nearly perfectly match the exact quantum results. This is the first time such a comparison has been done and it provides a strong validation of the Matsubara-dynamics theory.

Althorpe, Stuart
chemical dynamics, dissipative dynamics, Matsubara dynamics, path integrals, system-bath problem
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge
EPSRC (2112390)