Fluctuations in driven systems

•A formalism for describing systems in local thermal equilibrium is described.•Two building blocks, describing a drive and dissipation, are introduced and studied.•Their time evolution and the steady-state properties are described.•More complicated setups can be constructed from the building blocks.•The formalism is used to understand correlations in conserving transport channels.

The lecture notes give an introduction to some aspects of fluctuations in driven systems. First, a particularly straightforward derivation of fluctuation theorems and their relation to linear response will be given. Being particularly simple the derivation gives clear limitations on the regime of validity of the theorems. Second, an introduction to systems in local thermal equilibrium, in time and space, will be given. To this end, two building blocks will be studied. One a system which is driven by some cyclically changing external potential. The other consisting of two systems, at different temperatures, which are equilibrating. Both building blocks display an interesting time evolution. For example, the statistical properties of fluctuations of the energy in driven isolated systems generally fall into two types of distinct behaviors. Following this a systematic approach for describing more complicated systems will be outlined by studying other setups. Specifically, driven-dissipative systems and systems driven by two external baths at different temperatures. The formalism will be used to elucidate, in a minimal model, the origin and sign of correlation in a boundary driven system.