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The thermodynamics of the discrete nonlinear Schr\"odinger equation in thevicinity of infinite temperature is explicitly solved in the microcanonicalensemble by means of large-deviation techniques. A first-order phase transitionbetween a thermalized phase and a condensed (localized) one occurs at theinfinite-temperature line. Inequivalence between statistical ensemblescharacterizes the condensed phase, where the grand-canonical representationdoes not apply. The control over finite size corrections of the microcanonicalpartition function allows to design an experimental test of delocalizednegative-temperature states in lattices of cold atoms.

Condensation transition and ensemble inequivalence in the Discrete Nonlinear Schr\"odinger Equation