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In these four lectures I describe basic ideas and methods applicable to bothclassical and quantum systems displaying slow relaxation and non-equilibriumdynamics. The first half of these notes considers classical systems, and thesecond half, quantum systems. In Lecture 1, I briefly review the glasstransition problem as a paradigm of slow relaxation and dynamical arrest inclassical many-body systems. I discuss theoretical perspectives on how to thinkabout glasses, and in particular how to model them in terms of kineticallyconstrained dynamics. In Lecture 2, I describe how via large deviation methodsit is possible to define a statistical mechanics of trajectories which revealsthe dynamical phase structure of systems with complex relaxation such asglasses. Lecture 3 is about closed (i.e. isolated) many-body quantum systems. Ireview thermalisation and many-body localisation, and consider the possibilityof slow thermalisation and quantum non-ergodicity in the absence of disorder,thus connecting with some of the ideas of the first lecture. Lecture 4 is aboutopen quantum systems, that is, quantum systems interacting with an environment.I review the description of open quantum dynamics within the Markovianapproximation in terms of quantum master equations and stochastic quantumtrajectories, and explain how to extend the dynamical large deviation method tostudy the statistical properties of ensembles of quantum jump trajectories. Myoverall aim is to draw analogies between classical and quantum non-equilibriumand find connections in the way we think about problems in these areas.

physica a statistical mechanics and its applications

Aspects of non-equilibrium in classical and quantum systems: slow  relaxation and glasses, dynamical large deviations, quantum non-ergodicity,  and open quantum dynamics