Why is the movement of electrons random in electric field?

Question

Suppose an uncharged conductor is present isolated and there is no net electric field applied. Then by assumption we have the motion of electrons random,i.e. isotropic.

But now suppose there is a closed conductor present. We apply a net electric field and the electrons starts moving. Suppose steady state current is achieved.

We notice that there is has been created a velocity of electrons on the opposite direction of the applied field. I know that there is random movement of the electrons and they perform Brownian motion. My question is that is it biased in one direction because of the field?

Is the motion of elections in a particular direction more probable or they have more velocity in that direction?

Answer 1

Is the motion of elections in a particular direction more probable or they have more velocity in that direction?

Both are actually true in the Drude model. In presence of a field, the probability for the electron to move in the direction opposite to the field is enhanced. As a result, their Brownian motion is drifted, which results in an average velocity in the direction of the drift. However, the distribution of velocities will keep the same width because the Fokker-Planck equation is now $$\frac{\partial P}{\partial t}=-\langle\vec v\rangle\cdot\vec\nabla P+D\nabla^2P \tag 1$$ where $\langle\vec v\rangle$ is the average velocity resulting from the force. If the solution in absence of the field is $P(\vec r,\,t)$, then the solution of (1) is simply $P(\vec r-\langle\vec v\rangle t,\,t)$. This means that it corresponds to the zero-field solution in a reference frame performing a uniform translation with speed $\langle\vec v\rangle$.