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

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.

Figures (6)

• Figure 1: Two sample paths $X(t,\omega)$ of a stochastic process.
• Figure 2: Velocity profile for simple shear.
• Figure 3: Real $\chi'(\omega)$ and imaginary part $\chi"(\omega)$ of the complex susceptibility for a damped harmonic oscillator.
• Figure 4: Typical set-up of a scattering experiment. The sample is exposed to a plane-wave probe beam of particles or waves characterized by a wave vector $\boldsymbol{\mathrm{k}}_\textrm{i}$ and frequency $\omega_\textrm{i}$. The scattered particles in the direction of $\boldsymbol{\mathrm{k}}_\textrm{f}$ are collected at the detector covering a solid angle $\mathrm{d} \Omega_\textrm{f}$ and resolved according to their frequency $\omega_\textrm{f}$ or energy $\epsilon_{\textrm{f}}$.
• Figure 5: Which of these four functions corresponds to a correlation function? Top left: damped harmonic oscillator. Top right: simple relaxator. Bottom left: harmonic oscillator with negative damping. Bottom right: Gaussian coupling model.