# alternative derivation of Einstein relation

Is there a derivation of the Einstein–Smoluchowski relation without the assumption of the Boltzmann distribution? Every time I see a derivation, it always assumes the Boltzmann distribution, such as the derivation on wikipedia.

My problem is that the causality is upside down. For me, the Boltzmann distribution is caused by the dynamic motion of the atoms, not that the dynamic motion of the atoms is caused by the Boltzmann distribution. I have seen a claim here that it's a detailed balance, but even here they used the Boltzmann distribution assumption. I don't understand the Fluctuation-dissipation theorem, but there should be simple explanation for this.

A more microscopical method to get the relations for Brownian motion are Langevin equations (statistical differential equations). There you model the influence of the thermal bath by a stochastic force. In this model you can observe how the systems reaches thermal equilibrium from some initial conditions. The relation in question is reached in the formalism by assuming a time scale large compared to the damping time (which is equivalent to the assumption of thermal equilibrium). Langevin's original derivation also assumed the equilibrium by using the virial theorem in the form $\frac 1 2 m\langle \ddot x \rangle^2 = \frac 1 2 \langle T \rangle$.