Wigner distribution functions for complex dynamical systems: the emergence of the Wigner-Boltzmann equation

The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation

Variational Truncated Wigner Approximation

In this paper we reconsider the notion of an optimal effective Hamiltonian for the semiclassical propagation of the Wigner distribution in phase space. An explicit expression for the optimal effective Hamiltonian is obtained in the short time limit by minimizing the Hilbert-Schmidt distance between the semiclassical approximation and the real state of the system. The method is illustrated for the quartic oscillator

Occupation numbers in a quantum canonical ensemble: a projection operator approach

Recently, we have used a projection operator to fix the number of particles in a second quantization approach in order to deal with the canonical ensemble. Having been applied earlier to handle various problems in nuclear physics that involve fixed particle numbers, the projector formalism was extended to grant access as well to quantum-statistical averages in condensed matter physics, such as particle densities and correlation functions. In this light, the occupation numbers of the subsequent single-particle energy eigenstates are key quantities to be examined. The goal of this paper is 1) to provide a sound extension of the projector formalism directly addressing the occupation numbers as well as the chemical potential, and 2) to demonstrate how the emerging problems related to numerical instability for fermions can be resolved to obtain the canonical statistical quantities for both fermions and bosons.Comment: 23 pages, 8 figure

Self-energy correction to dynamic polaron response

We present the first order self-energy correction to the linear response coefficients of polaronic systems within the truncated phase space approach developed by the present authors. Due to the system-bath coupling, the external pertubation induces a retarded internal field which dynamically screens the external force. Whereas the effect on the mobility is of second order, dynamical properties such as the effective mass and the optical absorption are modified in first order. The Fr\"ohlich polaron is used to illustrate the results

Wigner distribution functions for complex dynamical systems: a path integral approach

Starting from Feynman's Lagrangian description of quantum mechanics, we propose a method to construct explicitly the propagator for the Wigner distribution function of a single system. For general quadratic Lagrangians, only the classical phase space trajectory is found to contribute to the propagator. Inspired by Feynman's and Vernon's influence functional theory we extend the method to calculate the propagator for the reduced Wigner function of a system of interest coupled to an external system. Explicit expressions are obtained when the external system consists of a set of independent harmonic oscillators. As an example we calculate the propagator for the reduced Wigner function associated with the Caldeira-Legett model