For systems in thermal equilibrium, it is well known from equilibrium statistical mechanics that fluctuations play important role, in particular in systems close to phase transitions. For non-equilibrium systems, fluctuations are
also important, giving rise to dynamical scalings, long-range correlations and capturing time reversal symmetry properties. In these two lectures we study a few aspects of non-equilibrium fluctuations by mainly treating one-dimensional systems which are analytically tractable.
We plan to cover mainly the two subjects. The first is the Kardar-Parisi-Zhang universality. We start by introducing basic models such as exclusion processes and KPZ equation. We then discuss the mapping to a problem of directed polymer in random media and exact solutions. We also discuss appearances of KPZ universality in other contexts, including anharmonic chains, random unitary circuit and quantum spin chains.
The second is the macroscopic fluctuation theory. This was introduced by a group of Jona-Lasinio et al around 2000 and has been developed since then. It is believed to describe large deviation aspects of non-equilibrium systems. Recently a few exact solutions for the MFT equations have been achieved by finding connections to classical integrable systems. We explain both basic aspects and the recent progress about the theory.