Quantum free energy differences from non-equilibrium path integrals: I. Methods and numerical application

TL;DR

This paper introduces methods combining path integral representation and non-equilibrium fluctuation relations to compute quantum free energy differences, demonstrating their numerical application with regularization techniques.

Contribution

It develops a novel approach linking quantum free energy calculations with non-equilibrium work relations via path integrals and introduces regularization methods for practical computation.

Findings

Validates methods with a quartic double-well potential

Shows the applicability of fluctuation relations in quantum systems

Provides a smoothing procedure for work distribution functions

Abstract

The imaginary-time path integral representation of the canonical partition function of a quantum system and non-equilibrium work fluctuation relations are combined to yield methods for computing free energy differences in quantum systems using non-equilibrium processes. The path integral representation is isomorphic to the configurational partition function of a classical field theory, to which a natural but fictitious Hamiltonian dynamics is associated. It is shown that if this system is prepared in an equilibrium state, after which a control parameter in the fictitious Hamiltonian is changed in a finite time, then formally the Jarzynski non-equilibrium work relation and the Crooks fluctuation relation are shown to hold, where work is defined as the change in the energy as given by the fictitious Hamiltonian. Since the energy diverges for the classical field theory in canonical equilibrium, two regularization methods are introduced which limit the number of degrees of freedom to be finite. The numerical applicability of the methods is demonstrated for a quartic double-well potential with varying asymmetry. A general parameter-free smoothing procedure for the work distribution functions is useful in this context.