This is the collection of summaries in the winter school (Winter school on Statistical Mechanics, Nonequilibrium Processes and Probability) in Rome in 2025.
Speaker: Eric Carlen

Title Quantum Markov Semigroups in Non-equilibrium Statistical Physics

Day 1: Large N limit of finite systems of N degrees of freedom. Quantum dynamical system Question: which generators of QDS on $T(H)$ arising from the coupling of quantum system on $H$ to a reservoir on some limit?

  1. What is the structure of the general generator of a quantum dynamical semigroups on $T(H)$ with $H$ finite dimensional Hilbert space?
  2. Steady state of generator $L^\dagger$ of a QDS
  3. When can we prove \(D(\rho_t^\dagger| \sigma) \le e^{-ct} D(\rho | \sigma)\) with an explicit $c$ under detailed balance condition with respect to the steady state $\sigma$.

Definition of completely positive, superoperator…

Follow up check: proof of convergence of equilibrium to Kac Boltzmann model, Gradient flow for Boltzmann-Kac equation, random walk on graph, qunantum binary collision model