# Why does the Einstein relation hold in the derivation of the Goldman equation?

The Einstein relation $$D = \mu k_B T$$ is derived by assuming an equilibrium between the drift current and the diffusion current. Knowing this I would assume, that the relation is only valid under this assumption.

However, when deriving the Goldman equation, we start with a combination of drift current and diffusion current: $$j_v = -D_v \frac{\partial c_v}{\partial x}+c_vv_{Drift}$$ and then use the Einstein relation to transfer this equation to the more complex version of the current $$j_v$$ which is stated on Wikipedia.

Why can we use the Einstein relation in this case, when drift and diffusion current are not in equilibrium? (If they were, $$j_v$$ would be 0, which is not the case).

The voltage driven drift and the concentration-driven diffusion are distinct physical effects so it is not double-counting to include both of them when computing a current flow. That there is a relation between them comes from the possibility of equilibrium in the situation of a static potential gradient. Einstein knew that in equilibrium the time-independent density $$\rho$$ would be proportional to the Boltzman factor $$\exp\{- V(x)/kT\}$$. This could only hold when the mobility and the diffusion constants obeyed his relation so that the sum of the drift and disffusion currents is zero.
To be precise: suppose that $${\bf v}_{\rm drift} = -\mu \nabla V$$ and $${\bf j}_{\bf diff} = -D \nabla \rho$$. These phenomenological equation hold in all processes, equilibrium or non equilibrium. Now consider a particular case. This is the effect of a potential $$V$$. We know from thermodynamics that this will lead to an equilibrium distribution in which $$\rho = \rho_0 \exp\{-V/kT\}$$ The net current from diffusion and voltage induced drift is $${\bf j}_{\rm tot}={\bf j}_{\rm drift} + {\bf j}_{diffusion}\\ = \rho {\bf v}_{\rm drift}+ D \rho_0 \exp\{-V/kT\}\frac 1{kT} \nabla V\\ = -\rho \mu \nabla V +D\rho \frac 1{kT} \nabla V$$ Since we are in equilibrium $${\rm j}_{\rm tot}=0$$. This can only hold for arbitrary $$V$$ if $$D=\mu kT$$.