Mesoscale mode coupling theory for the weakly asymmetric simple exclusion process

The asymmetric simple exclusion process and its analysis by mode coupling theory (MCT) is reviewed. To treat the weakly asymmetric case at large space scale $x{\epsilon }^{-1}$, %(corresponding to small Fourier momentum at scale $p\epsilon$), large time scale $t{\epsilon }^{-\chi }$ and weak hopping bias $b{\epsilon }^{\kappa }$ in the limit $\epsilon \to 0$ we develop a mesoscale MCT that allows for studying the crossover at $\kappa =1\mathrm{/}2$ and $\chi =2$ from Kardar-Parisi-Zhang (KPZ) to Edwards-Wilkinson (EW) universality. The dynamical structure function is shown to satisfy for all $\kappa$ an integral equation that is independent of the microscopic model parameters and has a solution that yields a scale-invariant function with the KPZ dynamical exponent $z=3\mathrm{/}2$ at scale $\chi =3\mathrm{/}2+\kappa$ for $0\le \kappa <1\mathrm{/}2$ and for $\chi =2$ the exact Gaussian EW solution with $z=2$ for $\kappa >1\mathrm{/}2$. At the crossover point it is a function of both scaling variables which converges at macroscopic scale to the conventional MCT approximation of KPZ universality for $\kappa <1\mathrm{/}2$. This fluctuation pattern confirms long-standing conjectures for $\kappa \le 1\mathrm{/}2$ and is in agreement with mathematically rigorous results for $\kappa >1\mathrm{/}2$ despite the numerous uncontrolled approximations on which MCT is based.