Dissipation and fluctuations are fundamental processes of macroscopic systems. If a system is driven out of equilibrium, it relaxes back towards the equilibrium state by dissipating the excess energy to the environment. If a constant force is applied to the system, a stationary non-equilibrium state is reached in which the energy gain due to the external force and the energy loss due to dissipation cancel. Also fluctuations are omnipresent both in equilibrium and non-equilibrium systems. In a canonical ensemble, for example, the energy fluctuates due to the energy exchange with a thermal bath. Interestingly it turns out that dissipation and fluctuations are intimately related. The connection between both is called the fluctuation-dissipation theorem.

In this course we will discuss two different approaches to treat dissipation and fluctuations in physical systems. The first one is Linear Response Theory. This is a very general approach to study the response of an arbitrary quantum mechanical system (but also applicable to classical systems) weakly driven out of equilibrium by an external force. The second one is the concept of Stochastic Processes, in which the randomness of events, caused, e.g., by the coupling to an external bath, is explicitly taken into account. This latter approach is also applicable to situations far from thermal equilibrium, but it is usually more closely related to a specific system and thus less general in its formulation.