# 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.

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).

• You said that "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)." I can't really understand it. What does the "duration" mean? – Cuckoo Sep 11 '14 at 6:40
• "Duration" is not a technical jargon term, it's just a word that means "length of time". – Steve Byrnes Sep 11 '14 at 20:47
• Hi Steve, I really feel like your answer has something deep and awesome that I should be getting, but I still don't understand. If I go to the dentist once every year or so (the electrons collide on average every $\tau$). If you were to ask me at a random time in the year in about how long I'll go to the dentist, on average I'd answer that I'll go in a half a year (A). This will be equal to my (average) answer to the question: "How long has it been since you last went to the dentist (B)". However, if you knew that I went to the dentist yesterday, and you came and asked me: – Joshua Ronis Mar 16 at 13:30
• "In about how long will you go to the dentist" - my answer will be: "In around a year" (C). And if you were to ask me: "When was the last time you went to the dentist" I'll answer that it was about a year ago (D). Those answers differ by a factor of 2. I know that what I'm missing between me going to the dentist on a consistent basis and the electrons crashing into atoms is that the electrons don't crash on a consistent basis. But I'm not sure how to intuitively explain it to myself Thanks! – Joshua Ronis Mar 16 at 13:34