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 $λ= \frac{1}{2}\frac{d^2}{dρ^2} χ(ρ) D(ρ)$ for the transport coefficient $λ$ of the KPZ equation, in terms of the conserved quantity $ρ$, the diffusion coefficient $D$ and the static compressibility of the system $χ$.

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