= Drude current relaxation
{c}
{title2=$\mathbf J(t)=\sigma_0\mathbf E_0+[\mathbf J(0)-\sigma_0\mathbf E_0]e^{-t/\tau}$}
Switch on a constant <electric field> $\mathbf E_0$ at $t=0$. For prescribed initial <current density> $\mathbf J(0)$, the exact <Drude model> solution is the displayed formula. For <Electrons>, the corresponding <drift velocity> is
$$
\mathbf v_d(t)=-\frac{e\tau}{m}\mathbf E_0+
\left[\mathbf v_d(0)+\frac{e\tau}{m}\mathbf E_0\right]e^{-t/\tau}.
$$
Starting from zero <drift velocity>, the <current density> reaches $1-e^{-1}\approx0.632$ of its limiting value after one <momentum relaxation time>. Switching the <electric field> off at $T$ instead gives $\mathbf J(t)=\mathbf J(T)e^{-(t-T)/\tau}$ for $t\geq T$. A finite step of <electric field> changes the acceleration immediately but leaves <drift velocity> and <current density> continuous.
\Image[/drude-electron-drift-current-transient.png]
{title=Exact <Drude current relaxation> during and after an <electric field> pulse}
{description=The applied <electric field> is $E_0\widehat{\mathbf x}$ for $0<t<3\tau$ and zero otherwise, with initially zero <current density>. The blue curve is $J_x/(\sigma_0E_0)$; the orange curve is $v_{d,x}/v_*$ for <Electrons>, where $v_*=eE_0\tau/m>0$. The opposite signs implement <conventional current>. These are calculated model curves.}
{source=drude-electron-drift-current-transient.py}
More generally, multiplying the <current density> equation by an <integrating factor> gives
$$
\mathbf J(t)=e^{-(t-t_0)/\tau}\mathbf J(t_0)+\frac{nq^2}{m}\int_{t_0}^{t}e^{-(t-s)/\tau}\mathbf E(s)\,ds.
$$
This is the <causal response of a first-order relaxation equation>: only earlier <electric fields> enter the <integral>. It also shows why instantaneous <Ohm's law> is an approximation when the <electric field> changes appreciably during a <momentum relaxation time>.
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