The Einstein relation for the KPZ equation

We compute the non-universal constants in the KPZ equation in one dimension, in terms of the thermodynamical quantities associated to the underlying microscopic dynamics. In particular, we derive the second-order Einstein relation $\lambda = \frac{1}{2}\frac{d^2}{d\rho^2} \chi(\rho) D(\rho)$ for the transport coefficient $\lambda$ of the KPZ equation, in terms of the conserved quantity $\rho$, the diffusion coefficient $D$ and the static compressibility of the system $\chi$.