  We propose a unified diffusion-mobility relation which quantifies both
quantum and classical levels of understanding on electron dynamics in ordered
and disordered materials. This attempt overcomes the inability of classical
Einstein relation (diffusion-mobility ratio) to explain the quantum behaviors,
conceptually well-settles the dimensional effect, phase transition and
nonlinear behavior of electronic transport. Our proposed theory relies on the
chemical potential which provides the coupling mechanism of charge-heat
current, due to electron-phonon coupling. We have derived expressions which
explain charge transport in both degenerate and nondegenerate materials, and
also provide the linear and nonlinear relationship between the charge density
and chemical potential. Theoretically, we find that the symmetrical nature of
electron-hole transport in strongly correlated two-dimensional semiconductors
indicates linear dispersion. We also observe the broken symmetry in the
nonlinear regime. This generalized diffusion-mobility relation explains both
the strongly and weakly correlated systems from low temperature to high
temperature, in both the relativistic as well as nonrelativistic domains. In
vanishing charge density limit of nondegenerate cases, the nonlinear transport
reduces to linear like transport, which is the classical Einstein relation.
