Roderich Tumulka - Eberhard Karls University Tuebingen
Wednesday, January 7, 2026
Seminar begins: 10:45AM EST
What really lies behind the grand-canonical ensemble in quantum mechanics
When textbooks introduce the grand-canonical ensemble rho=exp[-beta(H-mu_1 N_1-...-mu_r N_r)]/Z, they usually consider only systems defined by a volume in space without walls, so particles can enter or leave. The more important application would be systems for which particle numbers can change due to chemical reactions such as A+B=C+delta E with A,B,C three types of molecules and delta E the energy released during the reaction. Then for given delta E and initial particle numbers and (approximate) total energy, even the answer to the question how to compute the equilibrium particle numbers is not obvious, in part because it is not obvious which mu_i to use. Actually, the answer is easy to understand but perhaps not widely known. It makes use of a generalized Gibbs ensemble rho=exp[lambda_1 Q_1+...+lambda_K Q_K]/Z, where the operators Q_k mutually commute, and one of them is the Hamiltonian [Rigol et al. 2007 with precursor in Landau-Lifshitz 1980]. We formulate the general principle relevant to this ensemble and derive it for extensive Q_k. On top of that, an analog of canonical typicality applies, ensuring that for typical pure states with given values of the Q_k, small subsystems have density matrix given by this formula. And on top of that, one can predict not only the density matrix of a subsystem but even the probability distribution of its (conditional) wave function, which is related to the so-called GAP or Scrooge distribution over the unit sphere in Hilbert space. This is joint work with Cedric Igelspacher and Cornelia Vogel.