Momentum relaxation time ($\tau$)

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 $d⟨\mathbf{p}⟩/dt=q\mathbf{E}-⟨\mathbf{p}⟩/\tau$. This is a population equation: it does not assert that every particle stops at equally spaced instants.^[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 $ϵ/\sigma$: one describes loss of directed motion, the other redistribution of charge density.