Comparison of Matsubara dynamics with exact quantum dynamics for an oscillator coupled to a dissipative bath

Abstract

Matsubara dynamics is the classical dynamics which results when imaginary-time path-integrals are smoothed; it conserves the quantum Boltzmann distribution and appears in drastically approximated form in path-integral dynamics methods such as (thermostatted) ring-polymer molecular dynamics (T)RPMD and centroid molecular dynamics (CMD). However, it has never been compared directly with exact quantum dynamics for non-linear operators, because the difficulty of treating the phase has limited the number of Matsubara modes to fewer than 10. Here, we treat up to $\sim$200 Matsubara modes in simulations of a Morse oscillator coupled to a dissipative bath of harmonic oscillators. This is done by expressing the Matsubara equations of motion in the form of a generalised Langevin equation, approximating the noise to be real, and analytically continuing the momenta to convert the Matsubara phase into ring-polymer springs. The resulting equations of motion are stable up to a maximum value of modes which increases with bath coupling strength and decreases with system anharmonicity. The dynamics of the tail of highly oscillatory Matsubara modes is found to be harmonic, and can thus be computed efficiently. For a moderately anharmonic oscillator with a strong but subcritical coupling to the bath, the Matsubara simulations yield non-linear $⟨{\hat{q}}^2{\hat{q}}^2(t)⟩$ time-correlation functions in almost perfect agreement with the exact quantum results. Reasonable agreement is also obtained for weaker coupling strengths, where errors arise because of the real-noise approximation. These results give strong evidence that Matsubara dynamics correctly explains how classical dynamics arises in quantum systems which are in thermal equilibrium.

References in corpus (12)

• Competing quantum effects in the dynamics of a flexible water model
• Perspective: Numerically "exact" approach to open quantum dynamics: The hierarchical equations of motion (HEOM)
• Efficient stochastic thermostatting of path integral molecular dynamics
• How to remove the spurious resonances from ring polymer molecular dynamics
• Boltzmann-conserving classical dynamics in quantum time-correlation functions: Matsubara dynamics
• Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics
• Quantum dynamics using path integral coarse-graining
• Fast quasi-centroid molecular dynamics
• Testing the quasicentroid molecular dynamics method on gas-phase ammonia
• Dissipative Tunneling Rates through the Incorporation of First-Principles Electronic Friction in Instanton Rate Theory I: Theory
• Dissipative Tunneling Rates through the Incorporation of First-Principles Electronic Friction in Instanton Rate Theory II: Benchmarks and Applications
• Improved torque estimator for condensed-phase quasicentroid molecular dynamics

Cited by in corpus (1)

• Semiclassical Dynamics in Wigner Phase Space I : Adiabatic Hybrid Wigner Dynamics