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arXiv:1207.6909 · Physica A Statistical Mechanics and its Applications

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

Dries Sels, F. Brosens, Wim Magnus

Published Sep 14, 2012• 15 citations
arXivHTMLPDF
DOI
10.1016/j.physa.2012.09.007

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The Journal of Chemical Physics•Mar 15, 1985•14 citations
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.
The dynamic structure factor (DSF) is a mathematical function that contains information about inter-particle correlations and their time evolution. Mostly the classical molecular dynamics is used to calculate the DSF of the classical systems. On the contrary this article deals with quantum systems and the quantum dynamic structure factor. The Wigner formulation of quantum mechanics was used to derive the path integral representation of the DSF, which is based on the Wiener-Khinchin theorem showing relation of the the power spectrum of a random paths to their correlation function. The $3 {\rm D} $ quantum system of strongly correlated soft-sphere fermions was considered as an interesting physical example. The developed Wigner path integral Monte Carlo (WPIMC) approach has been developed to calculate the spin--resolved DSFs, the radial distribution functions (RDFs) and other thermodynamic functions in a wide range of density and temperatures. The physical meaning of the peaks arising on the RDFs and DSF have been analyzed and explained by the manifestation of the interference effects of the exchange and interparticle interactions and, as well as, the wave interference between multiple-scattering. This phenomenon in the system of the soft-sphere scatterers may to be the precursor effect of the Anderson localization, which finds its origin in the wave interference between multiple-scattering paths.
A new semiclassical theory is formulated by introducing a new distribution function to be propagated in phase space. Primarily this theory is intended to describe molecular collisions such as reactive scattering, but is also applicable to intramolecular processes including bound state problems. This new distribution function is a kind of characteristic function in the terminology of the phase-space-quantum theory developed by Weyl, Wigner, Moyal and others, and is called a dynamical characteristic function (DCF). The equation of motion and some properties of the DCF are presented. The essential feature of our theory is that the semiclassical DCF can be propagated using classical trajectories. This distribution function carried information not only on the density, but also on the phase (action integral). In this sense this theory can be regarded as a kind of phase-space-path-integral formalism. The semiclassical approximation is analyzed from the viewpoint of path integrals. The Wigner distribution function is also discussed in the context of our analysis. As a theoretical application, the semiclassical DCF is shown to reproduce the expression of the periodic orbit theory for bound states. Also, as a simple numerical application, a propagation of the DCF on a nonquadratic (Morse) potential is presented.