Effective grand-canonical description of condensation in negative-temperature regimes

The observation of negative-temperature states in the localized phase of the Discrete Nonlinear Schr\"odinger (DNLS) equation has challenged statistical mechanics for a long time. For isolated systems, they can emerge as stationary extended states through a large-deviation mechanism occurring for finite sizes, while they are formally unstable in grand-canonical setups, being associated to an unlimited growth of the condensed fraction. Here, we show that negative-temperature states in open setups are metastable and their lifetime $\tau$ is exponentially long with the temperature, $\tau \approx \exp(\lambda |T|)$ (for $T<0$). A general expression for $\lambda$ is obtained in the case of a simplified stochastic model of non-interacting particles. In the DNLS model, the presence of an adiabatic invariant, makes $\lambda$ even larger because of the resulting freezing of the breather dynamics. This mechanism, based on the existence of two conservation laws, provides a new perspective over the statistical description of condensation processes.