Regularized Boltzmann-Gibbs statistics for a Brownian particle in a nonconfining field

We consider an overdamped Brownian particle subject to an asymptotically flat potential with a trap of depth U0 around the origin. When the temperature is small compared to the trap depth (ζ=kBT/U01), there exists a range of timescales over which physical observables remain practically constant. This range can be very long, of the order of the Arrhenius factor e1/ζ. For these quasiequilibrium states, the usual Boltzmann-Gibbs recipe does not work since the partition function is divergent due to the flatness of the potential at long distances. However, we show that the standard Boltzmann-Gibbs statistical framework and thermodynamic relations can still be applied through proper regularization. This can be a valuable tool for the analysis of metastability in the nonconfining potential fields that characterize a vast number of systems.