A Divergence-Free Wigner Transform of the Boltzmann Operator Based on an Effective Frequency Theory

The centroid effective frequency representation of path integrals as developed by Feynman and Kleinert was originally aimed at calculating partition functions and related quantities in the canonical ensemble. In its path integral formulation, only closed paths were relevant. This formulation has been used by the present authors in order to calculate the many-body Wigner function of the Boltzmann operator, which includes also open paths. This usage of the theory outside of the original intention can lead to mathematical divergence issues for potentials with barriers, particularly at low temperature. In the present paper, we modify the effective frequency theory of Feynman and Kleinert by also including open paths in its variational equations. In this way, a divergence-free approximation to the Boltzmann operator matrix elements is derived. This generalized version of Feynman and Kleinert's formulation is thus more robust and can be applied to all types of barriers at all temperatures. This new version is used to calculate the Wigner functions of the Boltzmann operator for a quartic oscillator and for a double well potential and both static and dynamic properties are studied at several temperatures. The new theory is found to be essentially as precise as the original one. Its advantage is that it will always deliver a well-defined, even if approximate, Wigner function, which can, for instance, be used for sampling initial conditions for molecular dynamics simulations. As will be discussed, the theory can be systematically improved by including higher-order Fourier modes into the nonquadratic part of the trial action.