terminology - Definition of mean free time in the Drude model

In the Drude model they derive a formule for the conductivity of a conductor. I wonder though how the main free time $\tau$ is defined in this formula. Wikipedia says that it is "the average time between subsequent collisions". But I have two possible interpretations of this:

1. the average time an electron travels before colliding (which it seems to imply).


2. The average amount of time electrons have been travelling at a given time $t$ ($\tau$ will be substantially smaller in this definition).

The first definition seems to me how they describe it, while the second definition seems to be implied by the formulas. I wonder the same for "mean free path", which seems to be analogous.

Answer

This is a really good question, with a mind-bending answer. Check this out:

(A) Pick a random electron at a random time. How long (on average) do I need to wait until the next time it collides?

(B) Pick a random electron at a random time. How long (on average) has it been since the last time it collided?

(C) Pick a random electron that just collided. How long (on average) do I need to wait until the next time it collides?

(D) Pick a random electron that just collided. How long (on average) has it been since the last time it collided?

The answer is that all four of these are the same. (All four of them are what's called "mean free time".) It's mind-bending because it seems like (C) should be the sum of (A) plus (B), not the same as (A) and (B). But if you think about it, (C) and (A) have to be the same, because the fact that the electron just collided - something that happened in the past - cannot give any information about the statistics of what will happen to that electron in the future. We are assuming that collisions occur randomly! Similarly, (B) and (D) have to be the same. (A) and (B) have to be the same, obviously, because of time-reversal symmetry, and likewise (C) and (D) have to be the same.

Here's something that might help you come to peace with this apparent paradox (the so-called "Inspection Paradox"): An average where you pick a random electron at a random time is different than an average where you pick a random collision-to-collision free trajectory. (The former is relevant for (A-B), the latter is relevant for (C-D).) Compared to the former, the latter gives disproportionate weight to very short free trajectories. In other words, the ensemble of collision-to-collision free trajectories that you look at in (C) and (D) are, on average, shorter in duration than the ensemble of collision-to-collision free trajectories that you look at in (A) and (B).