We consider a diatomic chain characterized by a cubic anharmonic potential. After diagonalizing the harmonic case, we study in the new canonical variables, the nonlinear interactions between the acoustical and optical branches of the dispersion relation. Using the wave turbulence approach, we formally derive two coupled wave kinetic equations, each describing the evolution of the wave action spectral density associated to each branch. An H-theorem shows that there exist an irreversible transfer of energy that leads to an equilibrium solution characterized by the equipartition of energy in the new variables. While in the monoatomic cubic chain, in the large box limit, the main nonlinear transfer mechanism is based on four-wave resonant interactions, the diatomic one is ruled by a three wave resonant process (two acoustical and one optical wave): thermalization happens on shorter time scale for the diatomic chain with respect to the standard chain. Resonances are possible only if the ratio between the heavy and light masses is less than 3. Numerical simulations of the deterministic equations support our theoretical findings.

We predict negative temperature states in the Discrete Nonlinear Schödinger (DNLS) equation as exact solutions of the associated Wave Kinetic equation. Within the wave kinetic approach, we define an entropy that results monotonic in time and reaches a stationary state, that is consistent with classical equilibrium statistical mechanics. We also perform a detailed analysis of the fluctuations of the actions at fixed wave numbers around their mean values. We give evidence that such fluctuations relax to their equilibrium behaviour on a shorter time scale than the one needed for the spectrum to reach the equilibrium state. Numerical simulations of the DNLS equation are shown to be in agreement with our theoretical results. The key ingredient for observing negative temperatures in lattices characterized by two invariants is the boundedness of the dispersion relation. THE WAVE KINETIC THEORY FOR THE DNLS EQUATIONThe DNLS equation reads i ψm + (ψ m+1 + ψ m−1 − 2ψ m ) + ν|ψ m | 2 ψ m = 0, (1)