For Brownian particles initially released from an asymptotically flat potential well, equilibrium cannot be reached and the standard Boltzmann-Gibbs distribution is nonnormalizable since the usual partition function is divergent. However, when the potential well is deep enough compared to the temperature of the thermal bath, the dynamical and thermodynamical observables of the system remain almost constant for long times. We see heuristically that, for these quasi-equilibrium states, the standard Boltzmann-Gibbs framework can still be applied, through proper regularization, allowing to calculate ensemble averages and estimate the duration of the nonnormalizable quasi-equilibrium states. The validity of this regularized BG statistics is shown using the eigenfunction expansion of the time-dependent solution of the associated Fokker-Planck equation to obtain a time-independent approximation. Fractional dynamics and other extensions will also be discussed.
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