Consequences

For the description of electrons with momentum in the presence of an external force one has to solve a simple transport equation of the type

This leads to a first understanding of some important phenomena:

DC electrical conductivity

Let be the force on an electron in a constant electrical field. Let denote the current density, the average electronic velocity, m the mass of the electron, n the density of electrons. Clearly . Then the transport equation gives

and therefore

 

where denotes the DC conductivity. Equation (3) represents Ohm's law.

Hall effect

By considering the Lorentz force in a constant magnetic field the transport equation also in principle explains the Hall effect; it predicts a vanishing magnetoresistance, i. e. the resistivity should be independent of a perpendicular magnetic field.

Specific heat

Applying the kinetic theory of gases to the assumed gas of electrons in a metal one gets an electronic contribution of

 

to the specific heat, where denotes Boltzmann's constant. The ions are assumed to be immobile and therefore cannot contribute to the specific heat.

Thermal conductivity

A rough analysis of the temperature gradient driven transport gives a thermal conductivity

 

where is the mean square electronic speed.

Wiedemann-Franz law

Dividing equation (5) by equation (3) and assuming (4) and (all kinetic energy is thermal energy) one gets the Wiedemann-Franz law

 

with coefficient of the correct order of magnitude, but about half the typical value.

Maxwell-Boltzmann-Distribution

The statistics of identical but distinguishable particles with dispersion

 

is described by the velocity distribution