Fundamental problems in Statistical Physics XIV: Lecture on Correlation and response functions in statistical physics

TL;DR

This paper introduces correlation and response functions in statistical physics, emphasizing their fundamental role, mathematical characterization, and the importance of rigorous structural properties beyond standard physics curricula.

Contribution

It provides a rigorous overview of correlation and response functions, highlighting their mathematical foundations and structural properties often overlooked in physics education.

Findings

Correlation functions are characterized by Bochner's theorem.

Response functions are represented by Herglotz-Nevanlinna measures.

The paper discusses general structural properties of these functions.

Abstract

In the first part of these short lecture notes, we will present an introduction on (auto-)correlation functions and linear-response functions in the language of a physicist. In particular, the fluctuation-dissipation theorem in classical physics is presented underlining the central role of correlation functions. The fundamental importance of (auto-)correlation functions raises the natural question on how they are characterized in general without referring to the concrete underlying dynamical laws. Perhaps unexpectedly -- despite being elegant and long established in the mathematical literature (Bochner's theorem for correlations; Herglotz-Nevanlinna representations for response) -- this answer is not widely appreciated in physics, partly because the requisite tools lie outside the standard curriculum. In the second part we adopt a more rigorous viewpoint: we state the key structural properties of correlation functions and provide selected derivations of these results. Finally, we return to linear response and discuss general characterization results for response functions.