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Predicting whether an observable will dynamically evolve to thermalequilibrium in an isolated quantum system is an important open problem, as itdetermines the applicability of thermodynamics and statistical mechanics. TheObservable Thermalization framework has been proposed as a solution,characterizing observables that thermalize using an observable-specific maximumentropy principle. In this paper, we achieve three results. First, we confirmthe dynamical relaxation of local observables towards maximum entropy, in a 1DIsing chain. Second, we provide the most general solution to the maximizationproblem and numerically verify some general predictions about equilibriumbehavior in the same model. Third, we explore the emergence and physicalmeaning of an observable-specific notion of energy. Our results marksignificant progress towards a fully predictive theory of thermalization inisolated quantum systems and open interesting questions aboutobservable-specific thermodynamic quantities.

Observable Thermalization: Theory, Numerical and Analytical Evidence