Local Harmonic Approximation to Quantum Mean Force Gibbs State

TL;DR

This paper reviews and extends the local harmonic approximation for calculating the mean force Gibbs state in quantum systems with system-bath interactions, providing error estimates and applications to complex potentials and biological problems.

Contribution

It introduces an improved local harmonic approximation method with error bounds for quantum mean force Gibbs states, applicable to various interaction regimes and complex potentials.

Findings

Accurately estimates mean force Gibbs states in challenging regimes.

Shows the approximation reduces to known high-temperature and strong-coupling limits.

Finds significantly lower mutation probabilities in a DNA proton tunneling model.

Abstract

When the strength of interaction between a quantum system and bath is non-negligible, the equilibrium state can deviate from the Gibbs state. But the expression of such a mean force Gibbs state in an arbitrary parameter regime is unknown and is numerically challenging to determine. In this work, we first review the local harmonic approximation to this problem [Maier et al., Phys. Rev. E 81, 021107 (2010)], which can accurately determine the mean force Gibbs state when either the system-bath coupling or the temperature is large, or when the third and higher derivatives of the potential are small compared to certain system-bath specific parameters. In the appropriate limit, we show that the local harmonic approximation reduces to the ultra-strong coupling and high temperature results recently derived in the literature. After deriving an estimate for the error induced by this method, we apply it to study some systems, like a quartic oscillator and a particle in a quartic double-well potential. We also apply this method to analyze the proton tunneling problem in a DNA recently studied in literature [Slocombe et al., Comm. Phys., vol. 5, no. 1, p. 109, 2022], where our results suggest the equilibrium value of the probability of mutation to be orders of magnitude lower than the steady state value obtained there ($10^{-8}$ vs $10^{-4}$).