Week 05.02.2024 – 11.02.2024

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.

While traditional quantum mechanics focusses on systems conserving energy and probability, described by Hermitian Hamiltonians, in recent decades there has been ever growing interest in the use of non-Hermitian Hamiltonians. These can effectively describe loss and gain in a quantum system. In particular systems with a certain balance of loss and gain, PT-symmetric systems, have attracted considerable attention. The realisation of PT-symmetric quantum dynamics in optical systems has opened up a whole new field of investigations.

The properties of non-Hermitian quantum systems are often less intuitive than those of conventional Hermitian systems. Here we make use of the Husimi representation in phase space to analyse dynamical and spectral features. We consider the flow of the Husimi phase-space distribution in a semiclassical limit, leading to a first order partial differential equation, that helps illuminate the foundations of the full quantum evolution. Further, we demonstrate how ingredients of the dynamics can be used to construct approximate Husimi distributions of characteristic quantum states.