The Boltzmann transport equation (BTE) is based on classical Hamiltonian-statistical mechanics, as described in Section 2.4. The BTE gives the particle energy (in a system of particles) in terms of its position and momentum (x, p), and also allows for the determination of a nonequilibrium probability distribution of particles fi under an applied force (on a return to equilibrium, after an initial nonequilibrium state). These distributions are used in determining transport coefficients under the influence of driving forces in cases of local nonequilibria.
The Maxwell equations describe the propagation of EM waves and their interactions with electronic entities. These are among the most useful fundamental equations (including the laws/relations of Gauss, Faraday, Ampere, and Ohm).
In treating carriers as particles, the fluctuation–dissipation transport theory associated with Green and Kubo  gives general expressions for the transport coefficients, valid at all times and densities, in terms of correlation or autocorrelation functions calculated from a system at equilibrium. The kinetic-theory-based transport coefficients require, as observed in experiments, imposition of an external gradient. These gradients lead to a breakdown of the correlation. Computer simulations (i.e., many-body problems, such as MD simulations) have provided equilibrium results, which are used in the G–K fluctuation dissipation theory, as well as in nonequilibrium theory. The nonequilibrium (imposed gradient) simulations require extrapolation to large systems, as well as having the complications of steep gradients and boundary conditions.