A self-consistent criterion for the range of validity of weakly driven processes

TL;DR

This paper introduces a self-consistent criterion based on a typical length scale to determine the validity range of linear response theory for classical open systems, aiding in understanding when linear approximations are reliable.

Contribution

It proposes a novel, self-consistent criterion for linear response validity using a length scale derived from the fluctuation-response inequality, applicable to classical open systems.

Findings

The criterion is illustrated with Brownian particles in harmonic traps.

It is applied to systems exhibiting the Kibble-Zurek mechanism.

The physical meaning of the typical length is discussed from thermodynamic and information-theoretic perspectives.

Abstract

One of the longstanding open questions in linear response theory concerns its true range of validity. Determining when the linear approximation can be trusted typically requires knowledge of second-order corrections, which are often difficult to compute explicitly. In this letter, I propose a self-consistent criterion for the validity of linear response, formulated in terms of a typical length scale that emerges from the fluctuation-response inequality within the theory itself. The result applies to classical open systems. I illustrate the criterion with explicit examples of Brownian particles in harmonic traps, and classical open systems presenting Kibble-Zurek mechanism. Finally, I discuss the physical meaning of this typical length, providing both thermodynamic and information-theoretic interpretations.