Quantum free energy differences from non-equilibrium path integrals: II. Convergence properties for the harmonic oscillator

TL;DR

This paper analyzes the convergence of non-equilibrium path integral methods for quantum free energy differences in a harmonic oscillator, showing how different regularization techniques affect the rate of convergence.

Contribution

It establishes the convergence properties of work distribution and free energy calculations as the number of degrees of freedom increases, comparing Fourier and bead regularization methods.

Findings

Work distribution converges as M approaches infinity regardless of switching speed.

Fourier regularization leads to 1/M convergence, while bead regularization results in 1/M^2 convergence.

Results have implications for applying these methods to more complex quantum systems.

Abstract

Non-equilibrium path integral methods for computing quantum free energy differences are applied to a quantum particle trapped in a harmonic well of uniformly changing strength with the purpose of establishing the convergence properties of the work distribution and free energy as the number of degrees of freedom M in the regularized path integrals goes to infinity. The work distribution is found to converge when M tends to infinity regardless of the switching speed, leading to finite results for the free energy difference when the Jarzynski non-equilibrium work relation or the Crooks fluctuation relation are used. The nature of the convergence depends on the regularization method. For the Fourier method, the convergence of the free energy difference and work distribution go as 1/M, while both quantities converge as 1/M^2 when the bead regularization procedure is used. The implications of these results to more general systems are discussed.