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

Education innovation through material innovation in primary education : the grow-it-yourself workshop

In recent years more STEM (Science, Technology, Engineering and Mathematics) topics have been incorporated in mainstream public education. Although the benefits of STEM instruction are broadly recognised in secondary school curricula, STEM topics in primary education are rather limited, leaving a gap in manipulative skills building and in preparation processes for the next school level. This paper reflects on the outcomes of a design workshop attended by 12 primary school students (9 to 12 years old) in Belgium. Mycelium, a fungi-based natural material now used in innovative sustainable applications, served as a means to introduce early learners engineering basics through self-made learning tools. Students grew their own 3-D structures to build a 'Grow-It-Yourself biodegradable playground using mycelium as a primary source. The paper stems from an in-progress research that investigates the opportunities of how mycelium as a material innovation can be used as a medium to create innovation in primary education through a learning-by-design approach. Reflections on the workshop's instructional guidelines are included along with an extension of the call for support for primary school teachers delivering STEM topics in their classes

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

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

Thermodynamic fermion-boson symmetry in harmonic oscillator potentials

A remarkable thermodynamic fermion-boson symmetry is found for the canonical ensemble of ideal quantum gases in harmonic oscillator potentials of odd dimensions. The bosonic partition function is related to the fermionic one extended to negative temperatures, and vice versa.Comment: 7 pages, no figures, submitted to PHYSICA A. More information available at http://www.physik.uni-osnabrueck.de/makrosysteme

Confined Harmonically Interacting Spin-Polarized Fermions in a Magnetic Field: Thermodynamics

We investigate the combined influence of a magnetic field and a harmonic interparticle interaction on the thermodynamic properties of a finite number of spin polarized fermions in a confiment potential. This study is an extension using our path integral approach of symmetrized density matrices for identical particles. The thermodynamical properties are calculated for a three dimensional model of N harmonically interacting spin polarized fermions in a parabolic potential well in the presence of a magnetic field. The free energy and the internal energy are obtained for a limited number of particles. Deviations from the thermodynamical limit become negligible for about 100 or more particles, but even for a smaller number of fermions present in the well, scaling relations similar to those of the continuum approximation to the density of states are already satisfied.Comment: 7 pages REVTEX and 8 postscript figures, accepted in Phys. Rev.

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

Comment on: rotational properties of trapped bosons

Based on the Hellman-Feynman theorem it is shown that the average square radius of a cloud of interacting bosons in a parabolic well can be derived from their free energy. As an application, the temperature dependence of the moment of inertia of non-interacting bosons in a parabolic trap is determined as a function of the number of bosons. Well below the critical condensation temperature, the Bose-Einstein statistics are found to substantially reduce the moment of inertia of this system, as compared to a gas of ``distinguishable'' particles in a parabolic well.Comment: Herewith we repost our paper cond-mat/9611090 (1996). It was published in Phys. Rev. A 55, 2453 (March 1997), three years before cond-mat/0003471 (2000) by Schneider and Wallis. Reposted by [email protected]