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 ($\xi=k_B T/U_0 \ll 1$), 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 ${\rm e}^{1/\xi}$. For these quasi-equilibrium 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 (BG) 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 non-confining potential fields that characterize a vast number of systems.
Highlights
In nature, a system coupled to a thermal environment may not be confined indefinitely
When the temperature is small compared to the trap depth (ξ = kBT /U0 1), there exists a range of timescales over which physical observables remain practically constant
The partition function of the single particle is divergent, and we cannot apply the usual toolbox of equilibrium Boltzmann-Gibbs (BG) statistical mechanics even if the system appears to be in a thermal steady state [7,8,9,10,11]
Summary
A system coupled to a thermal environment may not be confined indefinitely. Another well-known example is when a Kramers reaction coordinate is in the vicinity of a metastable state [4,5,6], where the system can stay for a very long time, it is eventually destined to escape In all these examples, the partition function of the single particle is divergent, and we cannot apply the usual toolbox of equilibrium Boltzmann-Gibbs (BG) statistical mechanics even if the system appears to be in a thermal steady state [7,8,9,10,11].
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