Fermi pointed out that the Hydrogen atom in a thermal setting is unstable, as the
canonical partition function of this simple system diverges. We show how non-normalised
Boltzmann Gibbs measure can still yield statistical averages and thermodynamic proper-
ties of physical observables, exploiting a model of Langevin dynamics of a single Brownian
particle in an asymptotically flat potential . The ergodic theory of such systems is
known in mathematics as infinite (non-normalisable) ergodic theory, which in the second
part of the talk is discussed in the context of a gas of laser cooled atoms .
 E. Aghion, D. A. Kessler, and E. Barkai From Non-normalizable Boltzmann-
Gibbs statistics to infinite-ergodic theory Phys. Rev. Lett. 122, 010601 (2019)
 E. Barkai, G. Radons, and T. Akimoto Transitions in the ergodicity of subrecoil-
laser-cooled gases Phys. Rev. Lett. 127, 140605 (2021).