The Two Scaling Regimes of the Thermodynamic Uncertainty Relation for the KPZ-Equation

TL;DR

This paper analyzes the thermodynamic uncertainty relation for the 1+1 dimensional KPZ equation, revealing two distinct regimes separated by a critical coupling, with analytical and numerical results showing different asymptotic behaviors.

Contribution

It extends the understanding of the TUR for the KPZ equation to large coupling strengths and identifies two scaling regimes separated by a critical point.

Findings

TUR product approaches 5 in weak coupling limit

TUR product increases linearly with coupling in strong coupling regime

Analytical results agree with numerical simulations

Abstract

We investigate the thermodynamic uncertainty relation for the $(1+1)$ dimensional Kardar-Parisi-Zhang equation on a finite spatial interval. In particular, we extend the results for small coupling strengths obtained previously to large values of the coupling parameter. It will be shown that, due to the scaling behavior of the KPZ equation, the TUR product displays two distinct regimes which are separated by a critical value of an effective coupling parameter. The asymptotic behavior below and above the critical threshold is explored analytically. For small coupling, we determine this product perturbatively including the fourth order; for strong coupling we employ a dynamical renormalization group approach. Whereas the TUR product approaches a value of $5$ in the weak coupling limit, it asymptotically displays a linear increase with the coupling parameter for strong couplings. The analytical results are then compared to direct numerical simulations of the KPZ equation showing convincing agreement.