This chapter deals with correlation and response functions in equilibrium and in nonequilibrium steady states for a Langevin dynamics. First, the harmonic oscillator in equilibrium is discussed as a paradigmatic case. In the general nonlinear case, it is shown how time-derivatives in correlation functions can be replaced by state variables. The response function is derived within the path integral formalism. It can be expressed by various forms of a correlation function. One particularly transparent version restores the form of the equilibrium fluctuation-dissipation theorem for a nonequilibrium steady state. A second strategy to derive a response function starts with the perturbed Fokker–Plank operator. Causality imposes the Kramers–Kronig relations between the real and imaginary parts of the response function. Through the Harada–Sasa relation, the deviation from the equilibrium form of the fluctuation-dissipation relation can be related to the mean entropy production.

In this chapter, the student learns how to perform certain classes of definite integrals using contour integration methods. Although the integration variable is real for most integrals of interest, such as the inverse Fourier transform, analysis of the integral is extended to complex values of the integration variable and theorems related to integrating around closed contours on the complex plane are used to solve classes of definite integrals. The key theorems include Cauchy’s theorem for integrating so-called analytic functions, Jordan’s lemma, and the residue theorem for the important case where inside a closed contour on the complex plane, the integrand has places called singularities at which the function is not well behaved. Contour integration is used to analyze and derive results for the constitutive laws of a material when the current response depends not just on current forcing but also on the history of the forcing. This topic is called delayed linear response, which is developed at length. Contour integration, when combined with Fourier transforms, provides the solution of various types of initial-value and boundary-value problems in infinite and semi infinite domains.

Covers linear response from the one-electron viewpoint, including causality and the Kramers–Kronig relation. It develops the Kubo conductivity formula with special reference to the quantum Hall effect. The longitudinal and transverse dielectric functions are derived, and the ideas of intraband and interband, both direct and indirect, optical transitions are discussed.