# The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class

## TL;DR

This paper demonstrates that the space derivative of the KPZ equation, known as the noisy Burgers equation, exhibits Gaussian fluctuations and belongs to a different universality class than KPZ, which shows Tracy-Widom statistics.

## Contribution

It reveals that the derivative of the KPZ equation does not share the same universality class as KPZ, contrasting previous assumptions of their equivalence.

## Key findings

- KPZ displays Tracy-Widom fluctuation statistics.
- The derivative of KPZ exhibits Gaussian fluctuation behavior.
- Numerical simulations and renormalization group analysis support these results.

## Abstract

The Kardar-Parisi-Zhang (KPZ) equation is a paradigmatic model of nonequilibrium low-dimensional systems with spatiotemporal scale invariance, recently highlighting universal behavior in fluctuation statistics. Its space derivative, namely the noisy Burgers equation, has played a very important role in its study, predating the formulation of the KPZ equation proper, and being frequently held as an equivalent system. We show that, while differences in the scaling exponents for the two equations are indeed due to a mere space derivative, the fluctuations behave in a remarkably different way: while KPZ displays Tracy-Widom statistics, its derivative displays Gaussian behavior, hence being in a different universality class. We reach this conclusion via direct numerical simulations of the equations, supported by a dynamic renormalization group analysis of field statistics.