Observable Statistical Mechanics: An Observable-Specific Approach to Equilibration and Thermalization

In this talk, I will introduce “Observable Statistical Mechanics”, a novel approach to understanding equilibration and thermalization in isolated quantum systems. Tackling this complex challenge, open for almost a century, is crucial for the foundations of statistical mechanics as well as for the development of quantum technologies. While standard approaches make statements about density matrices, Observable Statistical Mechanics shifts the focus to observables. This not only brings the theory closer to experimental reality, but also acknowledges that observables can exhibit significantly different behaviors. For instance, I will discuss recent work in Many-Body Localized systems showing that, while it is true that some local observables do not thermalize, others do, and thus we can apply this generalized version of statistical mechanics to characterize their equilibrium behavior. Indeed, Observable Statistical Mechanics has provided us with remarkably accurate predictions about observables at equilibrium, confirmed in a large-scale numerical study. This includes determining observables’ full equilibrium probability distribution of outcomes without accessing the energy eigenstates. These results are obtained by focusing on experimentally realistic initial states and observables, the latter representing measurements with few outcomes, i.e., highly degenerate observables. In fact, I will provide some heuristic arguments why we expect the framework’s predictions to hold for this class of observables. Furthermore, I will highlight a special family of observables dubbed “Hamiltonian Unbiased Observables” which were analytically shown to obey the Eigenstate Thermalization Hypothesis. In any Hilbert space whose dimension is a prime power, there exist an extensive number of such observables and they can be algorithmically constructed. In addition, I will also discuss recent progress in applying the framework to systems with non-commuting charges (non-Abelian symmetries) and present some connections to quantum thermodynamics. Finally, I will conclude by discussing some avenues for future work.