Resistivity dependance on temperature

Question

I'm just starting to study electricity and magnetism. The resistivity $\rho$ of a conductor is defined as

$$\rho = \frac {\mathbf E}{\mathbf j},$$

where $\mathbf E$ is the electric field and $\mathbf j = \sum_in_iq_i\mathbf v_i$ is the current density ($n$= concentration of carriers, $q$ their charges and $\mathbf v$ their velocities). I know that there's a linear approximation of $\rho(T)$, $T$ the temperature of the conductor, that says the resistivity rises when the temperature rises; but how from the definition of $\mathbf j$ can I say that an increase of the temperature means an increase in $\rho$?

I appreciate your thoughts.

Answer 1

You can not find the temperature dependence from this definition. You should look into the microscopic details of the process. For that, you should refer to the Boltzmann transport equation from statistical mechanics. You can find it in a textbook on Statistical Physics or Condensed Matter physics.

In short, at high temperatures the resistivity is dominated by electron-phonon scattering, which gives rise to the linear dependence (See here for example).