| Bemerkung |
Content: In physics, if we want to describe the time evolution of a gas, there are different possibilities to do so. One is to imagine that the gas consists of a lot of particles and describe the time evolution of the position and the velocity of each particle by Newton's law (microscopic description). This ansatz has the advantage that it is very exact. But it has the disadvantage that in a gas we have of the order of 10^13 particles. These are too much equations to solve with a computer. But in many cases, it is even not necessary to know all the positions and velocities of every single particle. Therefore another ansatz is to describe the time evolution of macroscopic quantities as density, mean velocity and temperature (macroscopic description). This ansatz has the advantage that now we have less equations, only equations for the density, the mean velocity and the temperature. But it is only an averaged description. So it does not take into account the individual effect of the interactions of the particles. So if something happens on the level of particles which influences the macroscopic behavior, this is not a good description anymore. Therefore, there is a third possibility introduced by Boltzmann. Here, one still uses an averaged description, so it is not necessary to follow each single particle, but it is still possible to take into account the effect of interactions (kinetic description), for example the Boltzmann equation. Such a description, for example is used in a plasma. In this lecture, we will consider
- connection of the microscopic, the kinetic and the macroscopic description and basic mathematical concepts how to get from one to another
- Examples of models for different applications (hard-spheres, plasma, aerosols)
- notions of solutions in the kinetic description and basic concepts to study existence of solutions in the kinetic description
- basic concepts to study the qualitative behavior of solutions and the convergence to equilibrium in the kinetic description
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eG