Effective Grand Canonical Description of Condensation in Negative-Temperature Regimes

The observation of negative-temperature states in the localized phase of the discrete nonlinear Schrödinger 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 𝜏 is exponentially long with the temperature, 𝜏 ≈exp⁡(𝜆⁢|𝑇|) (for 𝑇 <0). A general expression for 𝜆 is obtained in the case of a simplified stochastic model of noninteracting particles. In the discrete nonlinear Schrödinger model, the presence of an adiabatic invariant makes 𝜆 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.