Physical applications of infinite ergodic theory

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Physical applications of infinite ergodic theory

Eli Barkai

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 properties of physical observables, exploiting a model of Langevin dynamics of a single Brownian particle in an asymptotically flat potential [1]. The ergodic theory of such systems is known in  mathematics as infinite (non-normalisable) ergodic theory, a framework that is widely applicable from laser cooled gases to non-linear systems.

[1] E. Aghion, D. A. Kessler, and E. Barkai , Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory, Phys. Rev. Lett. 122, 010601 (2019)