The Einstein relation for the KPZ equation

TL;DR

This paper derives the second-order Einstein relation for the KPZ equation in one dimension, connecting the transport coefficient to thermodynamic quantities of the microscopic dynamics.

Contribution

It provides a rigorous derivation of the Einstein relation for the KPZ equation, linking macroscopic transport coefficients to microscopic thermodynamic parameters.

Findings

Derivation of the second-order Einstein relation for KPZ

Explicit expression of the transport coefficient in terms of thermodynamic quantities

Connection between microscopic dynamics and macroscopic KPZ parameters

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 $\lambda = \frac{1}{2}\frac{d^2}{d\rho^2} \chi(\rho) D(\rho)$ 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 $\chi$.