= Drude model
{c}
The <Drude model> explains finite <electrical conductivity> by balancing <charge carrier> <acceleration> in an <electric field> against loss of directed <momentum> through <collisions>. Its useful surprise is that continuous microscopic motion can produce a constant macroscopic <current density>: the applied <electric field> continually restores the directed <momentum> lost to the material. The simplest version has one mobile species, constant <number density> $n$, <electric charge> $q$, <mass> $m>0$, and <momentum relaxation time> $\tau>0$, in a stationary, homogeneous, <isotropic> material. At zero <magnetic field>, its mean <drift velocity> and <current density> obey
$$
m\frac{d\mathbf v_d}{dt}=q\mathbf E(t)-\frac{m\mathbf v_d}{\tau},\qquad
\mathbf J=nq\mathbf v_d,\qquad
\tau\frac{d\mathbf J}{dt}+\mathbf J=\sigma_0\mathbf E(t),\qquad
\sigma_0=\frac{nq^2\tau}{m}.
$$
For <electrons>, $q=-e$ with positive <elementary charge> $e$, so $\sigma_0=ne^2\tau/m$. The <electron> <drift velocity> points against a constant <electric field>, whereas <conventional current> points along it. Random microscopic <velocities> can have zero mean without being zero individually; <drift velocity> is this population mean, rather than each particle's speed.<Lecture notes in Solid State 3: Classical free electron model for metals: The Drude model by Eytan Grosfeld>[pp. 2–3, Eqs. (1.2)–(1.11): collision averaging, electron force, drift and conductivity]{ref}
The same <first-order linear differential equation> predicts both <Drude current relaxation> after switching an <electric field> and the <Drude frequency response> to a sinusoidal <electric field>. Together they let a measured <electrical conductivity> be interpreted in terms of a density, a <mass>, and a time scale; a single constant-field measurement determines only their combination $nq^2\tau/m$.
The original 1900 paper <Zur Elektronentheorie der Metalle by Paul Drude> sought a common account of electrical transport and <heat transfer> using mobile charged particles and ideas from the <kinetic theory> of gases. Its introduction, especially printed pp. 568–569, explains that comparing different transport phenomena could constrain the populations of free and bound charged particles. The modern <momentum relaxation time> formulation above is a useful presentation of the model, not a verbatim translation of the original notation or its historical assumptions about <mass>.<Zur Elektronentheorie der Metalle by Paul Drude>[pp. 568–569: motivation for combining transport phenomena; OCR has spelling and formula errors]{ref}
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