The Karlsruhe Tritium Neutrino (KATRIN) experiment aims to determine the effective electron anti-neutrino mass with a sensitivity down to 0.2 eV/c$^2$ (90\% CL) through spectroscopy of gaseous tritium $\beta$-decay in the endpoint region. This challenging goal can only be reached through a precise examination of all systematic effects of the experiment. One of these effects is caused by a plasma in the highly luminous windowless gaseous tritium source. The plasma is generated by beta decay, subsequent partial ionization of the surrounding tritium gas. It produces an ab initio inhomogeneous potential throughout the source, which can change the shape of the measured electron spectrum.
The exterior experimental conditions generate unconventional plasma conditions resulting in a highly magnetized, partly collisional, multi-species, non-thermal (with thermal components), bound plasma. The combination of these properties makes an analytical description impossible. This thesis therefore focuses on the development and results of a new model of the plasma, using an iterative approach between the newly developed Monte Carlo code KARL and the particle in cell code ACRONYM.