Quantum and Classical Dynamics Seminar

Thermodynamics describes macroscopic systems using only a limited number of observables, yet quantum formulations often rely on knowledge of the full density matrix. In this talk, I will present a framework for quantum thermodynamics based on the information accessible through coarse-grained measurements. Starting from observational entropy, we introduce a generalized entropy associated with an algebra of observables, allowing us to account for genuinely quantum features such as non-commutativity. Within this framework, we derive a second law that captures irreversibility arising not only from interactions with the environment but also from internal equilibration under unitary dynamics. We also establish fluctuation theorems and introduce quantum macroscopic notions of work and heat, including an additional work contribution associated with changes in the accessible algebra, induced by external constraints or measurement backaction. This approach provides a unified description of macroscopic and stochastic thermodynamics in quantum systems, while requiring access to only a restricted set of observables.

In this talk, I will present a method of constructing periodic matrix-valued field whose associated elliptic operator forms a counterexample to the Bethe-Sommerfeld Conjecture (BSC) for two-dimensional reduced Maxwell operator. I will also discuss an application of this construction method on the topological properties of two-dimensional photonic systems. Using computer-assisted method, this construction extends to $\mathbb{R}^{3}$, providing a counterexample to the BSC of full three-dimensional Maxwell operators.

This work is co-authored with Habib Ammari (ETH Zürich), Yu Gao (HKUST) and assisted by ChatGPT 6 Astra. We also want to share our experience how AI tools help with mathematical research, and hope to gain more insights through discussions.