- #1
CAF123
Gold Member
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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.
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.