Proof of General Case
This is a proof in one dimension, but it is identical to a proof in two or three dimensions (just replace d/dx with ). Essentially the same proof is found in many places, for example see Kubo.
Suppose some potential energy U creates a force on the particle (for example, an electric force). We assume that the particle would respond, other things equal, by moving with velocity . Now assume that there are a large number of such particles, with local concentration as a function of position. After some time, equilibrium will be established: The particles will "pile up" around the areas with lowest U, but will still be spread out to some extent because of random diffusion. At this point, there is no net flow of particles: The tendency of particles to get pulled towards lower U (called the "drift current") is equal and opposite the tendency of particles to spread out due to diffusion (called the "diffusion current"). (See drift-diffusion equation.)
The net flow of particles due to the drift current alone is
(i.e. the number of particles flowing past a point is the particle concentration times the average velocity.)
The net flow of particles due to the diffusion current alone is, by Fick's laws
(the minus sign means that particles flow from higher concentration to lower).
Equilibrium requires:
In equilibrium, we can apply thermodynamics, in particular Boltzmann statistics, to infer that
where A is some constant related to the total number of particles. Therefore, by the chain rule,
Finally, plugging this in:
Since this equation must hold everywhere,
Read more about this topic: Einstein Relation (kinetic Theory)
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