The Einstein Relation for the KPZ Equation

Abstract

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 $$\lambda = \frac{a}{2}\frac{d^2}{d\rho ^2} \chi (\rho ) D(\rho )$$ for the transport coefficient $$\lambda $$ of the KPZ equation, in terms of the conserved quantity $$\rho $$ , the diffusion coefficient $$D$$ , the strength of the asymmetry $$a$$ and the static compressibility of the system $$\chi $$ .