Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation

TL;DR

This paper numerically tests the thermodynamic uncertainty relation for the KPZ equation, confirming analytical predictions in the weak coupling limit and exploring limitations due to discretization schemes.

Contribution

It provides a numerical validation of the thermodynamic uncertainty relation for the KPZ equation and analyzes discretization effects on entropy production accuracy.

Findings

Numerical results agree with analytical predictions in the weak coupling limit.

Discretization choices significantly affect the accuracy of entropy production estimates.

Intrinsic limitations to approximation accuracy are linked to the discretization scheme.

Abstract

A general framework for the field-theoretic thermodynamic uncertainty relation was recently proposed and illustrated with the $(1+1)$ dimensional Kardar-Parisi-Zhang equation. In the present paper, the analytical results obtained there in the weak coupling limit are tested via a direct numerical simulation of the KPZ equation with good agreement. The accuracy of the numerical results varies with the respective choice of discretization of the KPZ non-linearity. Whereas the numerical simulations strongly support the analytical predictions, an inherent limitation to the accuracy of the approximation to the total entropy production is found. In an analytical treatment of a generalized discretization of the KPZ non-linearity, the origin of this limitation is explained and shown to be an intrinsic property of the employed discretization scheme.