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We predict negative temperature states in the Discrete Nonlinear Sch\"odingerequation as exact solutions of the associated Wave Kinetic equation. Thosesolutions are consistent with the classical thermodynamics formalism. Explicitcalculation of the entropy as a function of the energy and number of particlesis performed analytically. Direct numerical simulations of the DNLS equationare in agreement with theoretical results. We show that the key ingredient forobserving negative temperatures in lattices is the boundedness of thedispersion relation in its domain. States with negative temperatures arecharacterized by an accumulation of particles and energy at wavenumber $k=\pi$.

Negative temperature states as exact equilibrium solutions of the Wave Kinetic equation for one dimensional lattices