Information-theoretic equilibrium and observable thermalization

TL;DR

This paper proposes a new observable-focused definition of thermal equilibrium in quantum systems, linking it to Shannon entropy maximization and the Eigenstate Thermalisation Hypothesis, with implications for understanding thermalization.

Contribution

It introduces an observable-centric approach to defining thermal equilibrium, connecting Shannon entropy maximization with quantum thermalization and providing a method to identify such observables.

Findings

Existence of observables with thermal equilibrium properties in closed quantum systems

Explicit construction method for equilibrium observables

Connection established between the new principle and Eigenstate Thermalisation Hypothesis

Abstract

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.