Recent advences on non-normalizable Boltzmann-Gibbs statistics and infinite-ergodic theory

The equilibrium state of a thermal system, in the presence of a strongly confining potential, is given by the famous Boltzmann-Gibbs distribution. This, along with the ergodic hypothesis, are hallmarks of statistical physics. If the potential is weakly confining, the Boltzmann factor is non-normalizable and the particle packet is constantly expanding. This gives rise to many questions. Among them: can we still infer the shape of the potential landscape, by observing the spatial distribution of the diffusing particles? How do we obtain ensemble and time-averaged observables in this case? And what is the entropy-energy relation in this system?

We show that the non-normalizable Boltzmann state is obtained from an entropy maximization principle, in the spirit of the canonical ensemble from standard thermodynamics, as well as from the ground state of the effective Hamiltonian of the system. The ergodic properties of important physical observables, like energy and occupation times in the system, are given by the Aaronson-Darlin-Kac theorem, and a generalized virial theorem is derived. Thus, our work extends the standard thermodynamic theory to a new class of potentials, and provides further strong evidence to the physical significance of infinite-ergodic theory.