= Momentum relaxation time
{title2=$\tau$}
The <momentum relaxation time> is the <exponential relaxation time> of mean directed <momentum> after the driving <electric field> is removed. In the simplest <Drude model>, independent <collisions> occur with probability $dt/\tau$ during a short <time> interval $dt$ and reset the mean <velocity> of the collided population to zero. The remaining fraction preserves its previous mean <momentum>. To first order in $dt$, averaging gives $d\langle\mathbf p\rangle/dt=q\mathbf E-\langle\mathbf p\rangle/\tau$. This is a population equation: it does not assert that every particle stops at equally spaced instants.<Lecture notes in Solid State 3: Classical free electron model for metals: The Drude model by Eytan Grosfeld>[pp. 2–3, Eqs. (1.2)–(1.4): constant collision probability and vanishing post-collision mean momentum]{ref}
For real materials, a <collision> that changes a particle's direction only slightly need not erase its directed <momentum>, so the <momentum relaxation time> need not equal the time between all scattering events. This transport time is also distinct from the bulk <charge relaxation> time $\epsilon/\sigma$: one describes loss of directed motion, the other redistribution of <charge density>.
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