Boltzmann equations are often used to describe the non-equilibriumtime-evolution of many-body systems in particle physics. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum-thermalization process, which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to full Kadanoff-Baym equations. Therefore, we performed a detailed comparison of Boltzmann and Kadanoff-Baym equations in the framework of relativistic quantum field theories in 3+1 space-time dimensions. In a first step, for simplicity we considered a real scalar Phi^4 quantum field theory and in a second step we generalized our results to a chirally invariant Yukawa-type quantum field theory including fermions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for Boltzmann equations. In this project we would like to continue our studies. The numerical solution of the above-mentioned Kadanoff-Baym equations requires a sophisticated optimization and parallelization of the algorithms as well as a corresponding high-performance computer hardware.The interaction of flows and magnetic fields in the electricallyconducting plasma of the convection zone and atmosphere of the Sun isthe origin of the various manifestations of solar activity. Of specialinterest in this connection is the formation of hydrodynamic structures(convective patterns, vortices) and of magnetic structures (flux tubes).As a result, the major part of the magnetic flux threading the solaratmosphere is concentrated in structures of large flux density (100 -300 mT) whose spatial distribution is determined by the convective flowpatterns. Magnetohydrodynamic waves and shocks propagating along themagnetic field contribute to heating the outer layers of the solaratmosphere and also to the acceleration of the solar wind. Thetheoretical description of these phenomena is based upon the equationsof magnetohydrodynamics and radiative transfer.
We carry out realistic three-dimensional time-dependent simulations ofsolar magneto-convection, aiming at approximating the real Sun, so thatthe results may be directly compared to observations. Therefore,elaborate physics is included: (non-local and frequency-dependent)radiative transfer, partial ionization (ionization equilibrium of the 11most abundant elements), open and transmitting boundary conditions,spectral line and polarization diagnostics. A typical simulation runsfor 1-2 hours solar time encompasses a small part of the uppermostconvection zone und photosphere of the Sun with an extension of 1400 kmin the vertical direction and 6000 km in both horizontal directions. Thegrid resolution is at least 288x288x100 grid points. The physicalproblems to be addressed during our simulation project include theformation of magnetic and convective patterns on various horizonatlscales of the solar atmosphere and their temporal development, theemergence of magnetic flux from the solar interior, the energy transportin dark sunspots and the interaction of magnetic field and convection onother stars than the Sun.