**Speaker:** Christian Maes

The first day of the school is devoted to an overview of the field of nonequilibrium statistical physics. We present a short historic perspective, the statistical mechanics basis of nonequilibrium, and introduce the guiding principles that prevailed in nonequilibrium thermodynamics.

- Brief history of nonequilibrium thermodynamics and statistical physics
- Dynamical characterizations of the equilibrium condition (H-theorem, Master equation, gradient flow)
- Open systems and local detailed balance (entropy production measuring time-reversal breaking)
- Standard models of nonequilibrium (ASEP, zero range, run-and-tumble, coupled oscillators,...)

**Speaker:** Christian Maes

This lecture aims at giving an up to date vision on the theory of nonequilibrium statistical physics. We present formal definitions, establishing a clear starting point, and the typical applications using recent examples.

- Dynamical ensembles (Action for Markov diffusion and jump processes)
- Linear response theory and fluctuation-dissipation relation in and outside equilibrium
- Applications of response theory (Sutherland-Einstein relations, negative conductivities)
- Static and Dynamical fluctuation theory (physics contents)

**Speaker:** Pierre de Buyl

In this lecture, we cover the standard modeling of nonequilibrium systems with simulations, for the most part using the Langevin equation formalism.

The model systems that we study are of interest for the current literature in experimental and theoretical nonequilibrium physics. We aim at providing a good starting point for new research projects. Examples include: colloids controlled by laser tweezers, self-propelled particles (Active Brownian Particles, Ornstein-Uhlenbeck particles) and the master equation.

We finish by a brief overview of Molecular Dynamics, the most suitable tool for considering many-body dynamics of active particles.

We review the implementation of the simulations, including the choice of appropriate algorithms and observables and the further analysis of the numerical experiments.

The program includes explicit numerical simulation of not-too-demanding models and every participant is expected to bring a laptop.

- A short introduction to numerical simulations
- Set up a Langevin simulation
- Measuring nonequilibrium dynamics
- Particles driven by an external field
- Self-propelled particles
- Simulation of the master equation
- Overview of Molecular Dynamics simulation

**Speaker:** Wojciech De Roeck

These two days of lecture will cover three fields that form the core of today's research on quantum nonequilibrium theory, namely

- Eigenstate Thermalization Hypothesis (ETH)
- Many-Body Localization (MBL) and glassy behaviour
- Prethermalization, periodic driving and time crystals

The lecture will start the topics from scratch and are thus also relevant for students and researchers who are not immersed in these specialized topics already. There are no prerequisites except for basic quantum theory and statistical mechanics.

The ETH is a property of wavefunctions of large systems that implies thermodynamic behaviour (dissipation and relaxation) for these systems. Since its proposal by M. Srednicki, it has been confirmed numerically in a convincing way.

The Many-Body Localization (MBL) phenomenon is exhibited by some low-dimensional quantum systems. It means that these systems do not thermalize, nor transport. They are perfect insulators. MBL can be defined as a (robust) absence of ETH. There is currently a lot of debate about what systems really show MBL and where this is merely an approximation. In any case, glassy or MBL-like behaviour is fairly widespread and can be easily understood.

Aspects of periodic driving have received a lot of attention due to their role for engineering Hamiltonians in synthetic quantum matter. This actually relies on a long prethermalization plateau. Time crystals are systems where the time-translation invariance is spontaneously broken.