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

Abstract

We consider an overdamped Brownian particle subject to an asymptotically flat potential with a trap of depth U_{0} around the origin. When the temperature is small compared to the trap depth (ξ=k_{B}T/U_{0}≪1), 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 e^{1/ξ}. 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.