Microscopic Description of Current Derivation

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Listening to lecture today in my E&M class, the professor was giving us the derivation of the micro description of current.

[tex]
I=neAv_d
[/tex]

Somewhere along the line he comes to [itex]v_d=at[/itex] that "t" might be a tau, I can't decipher my scribble. He said he isn't really sure how they get that particular expression and, "When in doubt, we turn to Feynman…unfortunately Feynman too, seems to do a lot of arm-waiving in this area. I'm not quite sure how this expression arises."

I know it has to do with the drift velocity of a charge carrier and that it relates to temperature. Can someone lead me to a reference or explain how this comes about? Super curious.

Thanks.
 

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