  We study a particle immersed in a heat bath, in the presence of an external
force which decays at least as rapidly as $1/x$, for example a particle
interacting with a surface through a Lennard-Jones or a logarithmic potential.
As time increases, our system approaches a non-normalizable Boltzmann state. We
study observables, such as the energy, which are integrable with respect to
this asymptotic thermal state, calculating both time and ensemble averages. We
derive a useful canonical-like ensemble which is defined out of equilibrium,
using a maximum entropy principle, where the constraints are: normalization,
finite averaged energy and a mean-squared displacement which increases linearly
with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs
statistics, thus extending the scope of the latter while shedding new light on
the concept of ergodicity.
