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For those of you who have a reasonable grasp of relativity and electrodynamics, has anybody ever tried to calculate the electrostatic attraction between two current carrying wires due to the Lorentz contraction of the "electron cloud"?

Physics
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or "electron sea" or however you wanna think of it... the flow of electrons.
please elaborate, sounds interesting
Are you referring to the length contraction of the electrons due to their velocity relative to the velocity of light?

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Imagine I have two long, straight wires running parallel to one another, each carrying current the same current. From the perspective of a stationary proton embedded in wire number 1, the electric field that I see is a superposition of the field generated by the stationary protons and the field generated by the moving electrons in wire number 2, since the electrons in wire number 1 all cancel. Classically the electrons and protons in wire number 2 would cancel each other but because the electrons are flowing in a group, like a cloud, the aggregate structure appears length contracted as per the consequences of special relativity.
@ Dalvoron not the electrons themselves, but the aggregate "cloud" they form as they drift through the wire.
This then increases the apparent number density of the electrons, which creates a net electric field that can be felt by the stationary protons.
|dw:1323677130678:dw|Is this roughly what you mean is happening?
That's the electron cloud.
Something like that, but since it's a continuous stream of electrons, the apparent density goes up.
|dw:1323673819978:dw| like the above train with oddly misshapen boxcars.
I'm not sure it does. The wires themselves are stationary, so the electrons are the only things that are being contracted (from the proton's point of view). Even though they travel in a cloud, the overall structure should remain the same, since the electrons don't move.|dw:1323677578708:dw|