Skip to main content
Library homepage
Chemistry LibreTexts

11: Linear Response Theory

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Correlation functions provide a statistical description of the dynamics of molecular variables; however, it remains unclear how they are related to experimental observables. You have probably sensed this from the perspective that correlation functions are complex, and how can observables be complex? Also, correlation functions describe equilibrium dynamics, but from a realistic point of view, exerting external forces should move the system away from equilibrium. What happens as a result? These questions fall into the realm of nonequilibrium statistical mechanics, an area of active research for which formal theories are limited and approximation methods are the primary tool. Linear response theory is the primary approximation method, which describes the evolution away or toward equilibrium under perturbative conditions.

    • 11.1: Classical Linear Response Theory
      We will use linear response theory as a way of describing a real experimental observable and deal with a nonequilibrium system. We will show that when the changes are small away from equilibrium, the equilibrium fluctuations dictate the nonequilibrium response! Thus knowledge of equilibrium dynamics is useful in predicting the outcome of nonequilibrium processes.
    • 11.2: Quantum Linear Response Functions
      To develop a quantum description of the linear response function, we start by recognizing that the response of a system to an applied external agent is a problem we can solve in the interaction picture.
    • 11.3: The Response Function and Energy Absorption
      Let’s investigate the relationship between the linear response function and the absorption of energy from the external agent—in this case an electromagnetic field.
    • 11.4: Relaxation of a Prepared State
      The impulse response function R(t) describes the behavior of a system initially at equilibrium that is driven by an external field. Alternatively, we may need to describe the relaxation of a prepared state, in which we follow the return to equilibrium of a system initially held in a nonequilibrium state. This behavior is described by step response function.

    Contributors and Attributions

    This page titled 11: Linear Response Theory is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Andrei Tokmakoff via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.