Editing Drude model (section)

=== Thermopower ===
A generic temperature gradient when switched on in a thin bar will trigger a current of electrons towards the lower temperature side, given the experiments are done in an open circuit manner this current will accumulate on that side generating an electric field countering the electric current. This field is called thermoelectric field:
<math display="block">\mathbf{E} = Q \nabla T</math>
and {{math|''Q''}} is called thermopower. The estimates by Drude are a factor of 100 low given the direct dependency with the specific heat.
<math display="block">Q = - \frac{c_v}{3ne} = - \frac{k_{\rm B}}{2e} = 0.43 \times 10^{-4} \mathrm{~V/K} </math>
where the typical thermopowers at room temperature are 100 times smaller, of the order of microvolts.<ref group="Ashcroft & Mermin" name=":22">{{harvnb|Ashcroft|Mermin|1976|pp=25}}</ref>
{{math proof|title=Proof together with the Drude errors<ref group="Ashcroft & Mermin" name=":21">{{harvnb|Ashcroft|Mermin|1976|pp=24}}</ref>|proof=

From the simple one dimensional model
<math display="block">v_Q=\frac{1}{2}[v(x-v\tau)-v(x+v\tau)]=-v \tau \frac {dv}{dx}= - \tau \frac {d}{dx}\left(\frac{v^2}{2}\right)</math>
Expanding to 3 degrees of freedom <math>\langle v_x^2 \rangle = \frac{1}{3} \langle v^2 \rangle</math> 
<math display="block">\mathbf{v_Q}=- \frac {\tau}{6} \frac {dv^2}{dT} (\nabla T)</math>
The mean velocity due to the Electric field (given the equation of motion above at equilibrium)
<math display="block">\mathbf{v_E}=- \frac {e \mathbf{E} \tau}{m}</math>
To have a total current null <math>\mathbf{v_E} + \mathbf{v_Q} = 0</math> we have
<math display="block">Q = - \frac{1}{3e}\frac {d}{dT}\left(\frac{mv^2}{2}\right) = - \frac{c_v}{3ne}</math>
And as usual in the Drude case <math>c_v=\frac{3}{2}nk_{\rm B}</math> 
<math display="block">Q = - \frac{k_{\rm B}}{2e} = 0.43 \times 10^{-4}~\mathrm{V/K} </math>
where the typical thermopowers at room temperature are 100 times smaller of the order of microvolts.<ref group="Ashcroft & Mermin" name=":22">{{harvnb|Ashcroft|Mermin|1976|pp=25}}</ref>
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