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I'm reading about Drude theory in the book Solid State Physics by Ashcroft and Mermin. This book and most other sources I can find simply state that the probability $P$ of an electron "collision" in a time $dt$ is just $$P=\frac{dt}{\tau}$$ where $\tau$ is the relaxation time.

I'm having trouble understanding how this follows from the basic definition of probability (i.e. desired outcome / all outcomes).

The Feynman lecture on diffusion came very close to what I'm looking for by writing: $$P=\frac{N_{collided}}{N} =\frac{\left(\frac{Ndt}{\tau}\right)}{N}=\frac{dt}{\tau}$$ where $N$ is the total number of particles, $N_{collided}$ is the number of particles that collided within time interval $dt$

Still, once you cancel out the $N$, I can't see why $\frac{dt}{\tau}$ makes sense as a probability.

kdaquila
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  • isn't this related to the Poisson process? A Poisson process at rate λ can be viewed as the result of performing an independent Bernoulli trial with success probability $p =$λ$dt$ in each “infinitesimal” time interval of length $dt$ https://en.wikipedia.org/wiki/Poisson_point_process , see also: https://physics.stackexchange.com/q/283419/226902 https://physics.stackexchange.com/q/518531/226902 . Moreover, related question: https://physics.stackexchange.com/q/601296/226902 – Quillo Sep 26 '22 at 15:37
  • For the poisson process, see the notes: http://www.columbia.edu/~ks20/4106-18-Fall/Notes-PP.pdf – Quillo Sep 26 '22 at 15:48

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I think I found the answer to my own question.

$P=\frac{1}{\tau}dt$ is a function for computing probability values. Although the function definition has the form of a ratio, it is not a probability value itself, so it doesn't have the form (desired outcomes) / (all outcomes).

$P=\frac{1}{\tau}dt$ is essentially the cumulative distribution function (CDF) for the uniform probability density function (PDF) $f=\frac{1}{\tau}$.

$P=\frac{1}{\tau}dt$ makes sense as a CDF because the probability of a collision should scale linearly with the value of the time interval $dt$ that you observe. The longer you observe, the higher the probability of getting a collision. Notable values would be at: $$(dt=0) \rightarrow P=0 $$ $$(dt=\tau) \rightarrow P=1 $$

Having a constant function $f=\frac{1}{\tau}$ makes sense as a PDF in this case because the probability of a collision during a given time interval $dt$ starting at t=0 will be the same as for starting at a later time value.

kdaquila
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