Boltzmann-conserving classical dynamics in quantum time-correlation functions: Matsubara dynamics

TL;DR

This paper introduces Matsubara dynamics, a classical approach derived from quantum mechanics that conserves the quantum Boltzmann distribution and improves accuracy over traditional semiclassical methods in computing quantum time-correlation functions.

Contribution

The authors derive a new classical dynamics method, Matsubara dynamics, by truncating the quantum Liouvillian in a way that preserves the quantum Boltzmann distribution, improving upon standard LSC-IVR.

Findings

Matsubara dynamics conserves the quantum Boltzmann distribution.

It converges better than LSC-IVR with increasing modes.

Numerical tests show improved agreement with exact quantum results.

Abstract

We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or classical Wigner approximation) results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e. a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads $N \to \infty$, such that the lowest normal-mode frequencies take their Matsubara values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of $\hbar^2$ at $\hbar^0$ (which gives back the standard LSC-IVR approximation), but in the normal-mode derivatives corresponding to the lowest Matsubara frequencies. The resulting Matsubara dynamics is inherently classical (since all terms $\mathcal{O}\left(\hbar^{2}\right)$ disappear from the Matsubara Liouvillian in the limit $N \to \infty$), and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum time-correlation function converges with respect to the number of modes, and gives better agreement than LSC- IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.