Linear Response Theory. Brownian Motion

This chapter consists of two parts. The first part dwells on linear response theory and the fluctuation-dissipation theorem that we prove rigorously in the full quantum setting. As a preparation we discuss in detail the structure and properties of the thermal correlation functions in Quantum Statistics: the defining KMS boundary conditions, their analyticity and causal properties, and the Kramers-Kronig dispersion relations. Then we prove the theorem which relates the response of a quantum thermal system to variations of the external conditions with the fluctuations of the dual observables. In the second part we discuss the Brownian motion both for its fundamental importance in its own right and as an illustration of the fluctuation-dissipation theorem in a classical setting. The Brownian motion is described both in terms of the stochastic Langevin equation and from the viewpoint of the Fokker-Planck diffusion process. We digress on the path integral aspects of the relation between the diffusion partial differential equations and the stochastic differential equations, highlighting their relations to supersymmetry. We close the chapter with a short discussion of the stochastic version of Hamilton’s equations and the allied Kramers’ diffusion PDEs.