# Drude model relaxation time approximation

1. Oct 25, 2014

### CAF123

In the Drude model of the free electron gas to explain the conduction of a metal, the relaxation time approximation that the electron has a collision in an infinitesimal time interval $dt$is $dt/\tau$. It can be shown that the mean time between collisions is $tau$. If we choose an electron at random, the average time since the last collision is $\tau$ and the average time to the next collision is $\tau$. The average time between the last collision and the next collision can then be shown to be $2\tau$ .
My question is how does this agree with the fact that the mean time between collisions is in fact $\tau$?
So in my mind, $\tau$ is that time where we consider some interval $[0,L]$ say and divide the total number of collisions in that interval by the time taken for the particle to travel from $0$ to $L$.
I am thinking the 2$\tau$ comes about by considering the fact that we might have a collision at $0$ and then at $0 + \epsilon, |\epsilon| \ll L$ and then maybe the next one is not until $3L/4$ and then again at $3L/4 + \epsilon$. At random, it is more likely to see the electron in the intervals $[0+\epsilon, 3L/4]$ than it is in the intervals $[0, 0+\epsilon]$ or $[3L/4, 3L/4 + \epsilon]$. If we apply this to a more symmetric set up then maybe this could explain the $2\tau$, but I am not sure if this observation is helpful at all.
Many thanks.

2. Oct 26, 2014

### CAF123

Can anybody provide any help?

3. Oct 27, 2014

### Cybertib

No. The time delay between two collisions is defined by τ.
"Time before next collision" - "time since last collision" = τ. It's not more complex than that...