Bernard Shizgal

The research work involves basic investigations in statistical mechanics with applications to atmospheric and space scie nce, ion and electron transport, chemical kinetics including hot atom reactions, stochastic processes, and rarefied gas dynamics.

The basis for most of the work is the Boltzmann equation or Fokker-Planck equation for the distribution function or probability density function. An important aspect of the work is the development of new methods of solution of the transport equations. A combination of analytical and numerical techniques is employed. In recent years, a Quadrature Discretization Method (QDM) has been used with considerable success in the solution of kinetic theory as well as quantum mechanical problems.

The QDM of solution of the Boltzmann equation is employed to determine the distribution functions in a variety of physical situations. These include the velocity distribution of ions in suitable moderators; the relaxation of electrons in atomic and molecular systems; the reaction and relaxation of hot atoms; diffusion in bistable potentia ls and the escape of atomic and molecular species from the atmospheres of the planets.

Work is in progress towards an accurate solution of the Boltzmann equation for a planetary atmosphere including the escape of lighter constituents. The nonthermal escape of atomic and molecular species from the atmospheres of Earth, Mars and Venus is also considered. This research involves the theoretical modeling of the terrestrial exosphere and the coupling with the ionosphere-magnetosphere systems. A major objective will be the modeling of the exosphere-ionosphere system from the collision dominated region at low altitudes to the collisionless region at high altitudes. This includes the kinetic theory of energetic species, notably atomic oxygen and O+, and in particular the energization mechanisms that lead to escape. The project is one of several research projects in preparation for the launch of the Enhanced Polar Outflow Probe (ePOP) in 2011, and is carried out in collaboration with Professor Andrew Yau of the Institute of Space Imaging Science at the University of Calgary. A description of the satellite mission, science objectives and the science team can be found at http://epop.phys.ucalgary.ca/index.html. The theoretical modeling results will be compared with data from ePOP.

A number of physical situations such as reaction rates in solution, optical bistability in lasers, reaction diffusion equations in modelling cardiac arrhythmias and other diffusive-type processes are studied with the Kramers and/or Fokker-Planck equations. Efficient methods of solution of these equations have been developed.

Recent work includes the development of the Inverse Polynomial Reconstruction Method (IPRM) in the resolution of Gibbs phenomenon. This new method for the removal of spurious oscillations in Fourier representations will provide a new method for the reconstruction of images in tomography.