The quantitative description of the transport mechanisms in solid oxide fuel cells (SOFCs) is relevant for cell development and the optimization of operating strategies. These mechanisms have an intrinsic multi-causality as given by Non-Equilibrium Thermodynamics (NET), which is not necessarily considered by empirical classical transport equations. If, e.g., additional heat is transported due to a potential gradient across the SOFC, this effect may have to be considered for the design of heating or cooling strategies. The main focus of this thesis is the description of the transport mechanisms in the electrolyte of a SOFC. The electrolyte is an essential part of the cell and should be highly ionic conductive, gas-tight and an electronic insulator. The electrolyte materials analyzed are zirconium dioxide ZrO2 co-doped with yttrium(III) oxide Y2O3 (YSZ) and ZrO2 co-doped with 10 mol% scandium(III) oxide Sc2O3 and 1 mol% cerium dioxide CeO2 (10Sc1CeSZ). This work comprises three molecular dynamics (MD) studies, which provide data for the phenomenological coefficients based on NET for different YSZ compositions, as well as the ionic conductivity and different diffusion coefficients for 10Sc1CeSZ. The numerical data for the ionic conductivity of 10Sc1CeSZ agrees with experimental studies. Furthermore, the dependency of each transport mechanism on the Y2O3 concentration in YSZ is explained by linear response theory. A theoretical framework is proposed to give the electrostatic potential thermodynamic consistency and to relate it to the Coulomb contribution of other thermodynamic quantities. Finally, simulations of a planar SOFC with an YSZ electrolyte are carried out using a validated one-dimensional(1D) NET model and the phenomenological coefficients from the MD simulations. If the coupled mechanisms are neglected, the temperature profile, the heat flux and the entropy production may not be correctly predicted. The methodologies and results from this work can be used in future studies to describe more accurately the transport mechanisms in SOFCs with electrolytes based on ZrO2 metal oxides or even other electrolyte technologies like proton conducting perovskites. Moreover, they can also be used to effectively predict the thermal behavior of SOFCs and, thus, they provide a contribution to the optimization of thermal strategies in SOFCs.