# Field-Theoretic Thermodynamic Uncertainty Relation -- General formulation exemplified with the Kardar-Parisi-Zhang equation

## TL;DR

This paper develops a field-theoretic thermodynamic uncertainty relation applicable to non-linear Langevin equations, exemplified by the Kardar-Parisi-Zhang equation, and demonstrates its validity through perturbation analysis.

## Contribution

It introduces a general field-theoretic framework for thermodynamic uncertainty relations and applies it to the KPZ equation, extending previous discrete and Langevin-based results.

## Key findings

- The relation holds up to second order in perturbation expansion.
- The framework describes current, entropy, and diffusivity in field theories.
- Validated the relation for the KPZ equation with small non-linearity.

## Abstract

We introduce a field-theoretic thermodynamic uncertainty relation as an extension of the one derived so far for a Markovian dynamics on a discrete set of states and for overdamped Langevin equations. We first formulate a framework which describes quantities like current, entropy production and diffusivity in the case of a generic field theory. We will then apply this general setting to the one-dimensional Kardar-Parisi-Zhang equation, a paradigmatic example of a non-linear field-theoretic Langevin equation. In particular, we will treat the dimensionless Kardar-Parisi-Zhang equation with an effective coupling parameter measuring the strength of the non-linearity. It will be shown that the field-theoretic thermodynamic uncertainty relation holds up to second order in a perturbation expansion with respect to a small effective coupling constant.