#### Working Group 1 : Equilibration, thermalisation and emergence of canonical states

How do complex quantum systems come to equilibrium, once pushed out of it? On what timescale can one expect this to happen? And how do properties such as thermal states dynamically emerge from a microscopic quantum description?

While statistical physics has recipes for treating quantum systems in equilibrium, and how thermal states can be grasped as maximum entropy states given suitable constraints, a long-standing open question is to resolve the tension between a microscopic description and the one offered by equilibrium statistical mechanics. Some mechanisms, that are relevant for processes of equilibration and thermalisation, have been identified early on, yet key questions still remain wide open.

##### Current state of knowledge

Given that questions concerning equilibration and thermalisation in quantum theory were already posed in the twenties of the last century, there has been a long tradition of addressing them. Still, technical progress on these questions is often remarkably recent, to a large extent due to the advent of new theoretical methods and new theory work which has in turn been stimulated by unprecedented experimental work.

A significant body of recent substantial work on the subject has been provided by participants of this Action. This includes theory results on kinematic approaches and notions of typicality, proofs of equilibration of subsystems for most times, proofs of strong equilibration in classes of free models and randomization under random Hamiltonians.

Significant progress has also been made concerning when one can expect thermalisation in translationally invariant systems to happen; but it has also been shown that contrary to common belief, this is by no means generically true for non-integrable systems and the situation is much more intricate than previously believed.

Significant recent experimental work within the consortium includes the exact monitoring of equilibration, pre-thermalisation and thermalisation in strongly correlated lattice models using optical superlattices and continuous systems using atoms chips.

##### Goals of the Working Group

In general terms, equilibration is the process in which a system under a given dynamics reaches a steady state. Thermalisation studies whether this steady state can be interpreted as a thermal state at a given temperature.

The main scientific tasks of this WG are:

- Study under what conditions systems reach an equilibrium state and when this state has a thermal form. Survey regimes in which thermalisation provably holds, clarify how a system “forgets” all but the most basic properties of the initial state and identify and investigate regimes where systems provably do not thermalize, challenging widely accepted criteria such as the eigenstate thermalisation hypothesis or integrability.
- Use new tools from quantum information theory, such as measure concentration, random Hamiltonian sampling and decoupling theorems, to re-address fundamental issues related to the time scales of equilibration and thermalisation. Improve these tools for the purpose, for instance, find a suitably general, yet computationally manageable, class of Hamiltonians with controlled interaction strength instead of random sampling over all possible types of system-bath interaction Hamiltonians.
- Formulate general statements of how the structure of the system and of its coupling with the environment determine the thermalisation process and induce quantum correlations in the system, both in the weak and in the strong coupling limit. Closing the gap in understanding between the closed and open scenario will provide further insight into our understanding of decoherence in quantum systems, the quantum measurement problem and the emergence of a classical world.
- Establish which variables play an important role in non-equilibrium processes. Specifically, which variables suffice to describe the equilibrium/thermal state and how can one formulate a generally valid non-equilibrium definition of entropy or temperature? Relate this to whether a many-body quantum system with a set of conserved quantities can relax to an equilibrium state, and what are its properties.