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

Pierre Nazé

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.

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

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.
Paper Structure (13 sections, 25 equations, 2 figures)

This paper contains 13 sections, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weakly driven processes
  3. Fluctuation-response inequality
  4. Quasistatic limit
  5. Examples
  6. Moving trap: degenerate case
  7. Stiffening trap: non-degenerate case
  8. Kibble-Zurek mechanism in open systems: failure of linear response theory
  9. Fisher information geometry
  10. Final remarks
  11. Relative entropy under linear response theory
  12. Fluctuation-response inequality
  13. Quasistatic limit of relative entropy

Figures (2)

  • Figure 1: Matching of irreversible work curves for perturbation $\delta\lambda=\sqrt{2}\lambda_0/100$ for exact and linear response theory of the stiffening trap. It was used $\lambda_0=1$, $\beta=1$, and $\tau_R=1/2$ for the linear protocol $g(t)=t$.
  • Figure 2: Disagreement of irreversible work curves for exact and linear response theory for perturbation $\delta\lambda=\sqrt{2}\lambda_0$ for the stiffening trap. It was used $\lambda_0=1$, $\beta=1$, and $\tau_R=1/2$ for the linear protocol $g(t)=t$.