New equilibrium ensembles for isolated quantum systems

The unitary dynamics of isolated quantum systems does not allow a pure state to thermalize. Because of that, if an isolated quantum system equilibrates, it will do so to the predictions of the so-called "diagonal ensemble" $\rho_{DE}$. Building on the intuition provided by Jaynes' maximum entropy principle, in this paper we present a novel technique to generate progressively better approximations to $\rho_{DE}$. As an example, we write down a hierarchical set of ensembles which can be used to describe the equilibrium physics of small isolated quantum systems, going beyond the "thermal ansatz" of Gibbs ensembles.Comment: Paper written for the Special Issue "Thermalization in Isolated Quantum Systems" of the Journal Entrop

On the error estimate of gradient inclusions

The numerical analysis of gradient inclusions in a compact subset of $2\times 2$ diagonal matrices is studied. Assuming that the boundary conditions are reached after a finite number of laminations and using piecewise linear finite elements, we give a general error estimate in terms of the number of laminations and the mesh size. This is achieved by reduction results from compact to finite case.Comment: 21 pages, 4 figure

Information-theoretic equilibrium and observable thermalization

To understand under which conditions thermodynamics emerges from the microscopic dynamics is the ultimate goal of statistical mechanics. Despite the fact that the theory is more than 100 years old, we are still discussing its foundations and its regime of applicability. A point of crucial importance is the definition of the notion of thermal equilibrium, which is given as the state that maximises the von Neumann entropy. Here we argue that it is necessary to propose a new way of describing thermal equilibrium, focused on observables rather than on the full state of the quantum system. We characterise the notion of thermal equilibrium, for a given observable, via the maximisation of its Shannon entropy and highlight the thermal properties that such a principle heralds. The relation with Gibbs ensembles is brought to light. Furthermore, we apply such a notion of equilibrium to a closed quantum systems and prove that there is always a class of observables which exhibits thermal equilibrium properties and we give a recipe to explicitly construct them. Eventually, we bring to light an intimate connection of such a principle with the Eigenstate Thermalisation Hypothesis.Comment: Accepted by Scientific Repor

Relaxation and 3d-2d passage with determinant type constraints: an outline

We outline our work (see [1,2,3,4]) on relaxation and 3d-2d passage with determinant type constraints. Some open questions are addressed. This outline-paper comes as a companion to [5]