I can't find the Einstein relation for spin diffusivity

I'm studying the article "Thermodynamic analysis of interfacial transport and of the thermomagnetoelectric system" (PhysRevB.35.4959, Mark Johnson and R. H. Silsbee); they use this relation $$D=\frac{\sigma \beta^2}{\chi e^2}$$ calling it "Einstein relation", but I can't find it anywhere on textbooks nor online, where the actual Einstein relation is written in terms of the mobility and thermal energy $$D=\mu k_B T$$ What is the origin of that relation? Where I can find it?

I am not sure about the $$\beta^2$$ part. Coleman derives $$\sigma = e^2\chi D$$ in his book Introduction to many-body physics Sec 10.4. The short version of it is that the density-density response function receives a correction when there are impurities. In perturbation theory in momentum $$q$$, one can identify the coefficient of $$q^2$$ as the diffusion constant.
Consider applying a small field $$E_x$$ onto a 2DEG. Denote the changed Fermi energy at the left edge $$\mu_1$$, and $$\mu_2$$ at the right edge. To make the connection with diffusion, remember that in a Fermi liquid, currents are carried by electrons near the Fermi surface. In other words, the electrons that actually make contributions are the ones in the energy interval $$\mu_2. The net effect of the electric field is to produce a density gradient across the 2DEG. Using the diffusion equation $$J = -eD\nabla n = eD\chi(\mu_1-\mu_2)/L = e^2D\chi E_x$$, we get $$\sigma = e^2\chi D$$. $$\chi$$ here is the density of states.
One can also think the current as simply $$J = env_d$$, where $$n$$ is the electron density and $$v_d$$ is the drift velocity. Since mobility $$\mu = e\tau/m$$ (and $$v_d = eE_x\tau/m$$), conductivity $$\sigma = en\mu$$. Comparing with the expression above, we get the usual $$D=E_f\mu/e=v_f^2\tau/2$$.
Notice this picture only works for a Fermi liquid or a degenerate (semi)conductor. For a non-degenerate (semi)conductor, the important energy scale is $$k_BT$$ rather than $$E_f$$. Then we get the second relation you have: $$D = \mu k_BT/e$$.