  We present the exact analytical equation of diffusion-mobility for
two-dimensional (2D) Schr\"odinger type transport systems, from molecules to
materials. The density of electronic states in such Schr\"odinger systems
pertains to the 2D non-relativistic carrier dynamics. We implement the Gaussian
function into carrier density derivation; accordingly we develop the electronic
compressibility and diffusion-mobility for both the generic and the degenerate
Fermi systems. This model is originally developed from generalized Einstein
relation, along with concern about the thermodynamic effects on many-body
interactions. The effect of interactions is included through the imperfect
Fermi-gas entropy function. Our extended model explains the cooperative
behavior of thermal and electronic counterparts on diffusion-mobility in
disordered systems at wide temperature range. Using earlier experimental and
theoretical results, we have shown the validity of our extended Einstein model
for different 2D degenerate systems. The results validate the original Einstein
equation at certain sets of temperature and chemical potential values for
different Gaussian variances. Beyond those combinations, the deviation is
observed. At very low temperature, the diffusion-mobility depends only on
chemical potential, which is the extended Einstein equation for ideal quantum
materials.
