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

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 ${\hslash }^2$ at ${\hslash }^0$ (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 $\mathcal{O}\left({\hslash }^2\right)$ 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.

References in corpus (10)

• How to remove the spurious resonances from ring polymer molecular dynamics
• Mapping Variable Ring Polymer Molecular Dynamics: A Path-Integral Based Method for Nonadiabatic Processes
• Direct simulation of electron transfer using ring polymer molecular dynamics: Comparison with semiclassical instanton theory and exact quantum methods
• Boltzmann-conserving classical dynamics in quantum time-correlation functions: Matsubara dynamics
• Derivation of a true (t->0+) quantum transition-state theory. Part I: Uniqueness and equivalence to ring-polymer molecular dynamics transition-state-theory
• Kinetically Constrained Ring-Polymer Molecular Dynamics for Non-adiabatic Chemical Reactions
• Quantum effects in the diffusion of hydrogen on Ru(0001)
• Derivation of a true (t->0+) quantum transition-state theory. II. Recovery of the exact quantum rate in the absence of recrossing
• On the uniqueness of t->0+ quantum transition-state theory
• Quantum Transition-State Theory

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