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Video abstract for the article 'Discrete breathers and negative-temperature states' by S Iubini, R Franzosi, R Livi, G-L Oppo and A Politi (S Iubini et al 2013 New J. Phys. 15 023032). Read the full article in New Journal of http://iopscience.iop.org/1367-2630/15/2/023032/article. GENERAL SCIENTIFIC SUMMARY Introduction and background. Since the pioneering work of Purcell, Pound and Ramsey in quantum nuclear-spin systems in the 1950s, physical states at negative temperatures have attracted the curiosity of researchers and shown how science can challenge common sense. In negative-temperature regimes, the temperature is above infinity and high-energy states are more populated than low-energy ones. In spite of the many years that have elapsed since the first general claims, it is still unclear whether negative temperature states are well defined in thermodynamics and how one would introduce simple protocols to access them in feasible experimental conditions. Main results. We tackle both issues with reference to the discrete nonlinear Schrödinger equation, a model that is physically relevant in the context of Bose--Einstein condensation in optical lattices and of light propagation in arrays of optical waveguides. By monitoring the micro-canonical temperature, we show that there exists a parameter region where the system evolves towards a state characterized by a finite density of discrete breathers and a negative temperature. Such a state persists over extremely long (astronomical) times since the convergence to equilibrium becomes increasingly slow as a consequence of a coarsening process. Discrete breathers are spatially localized modes of excitations corresponding to high atomic densities or high light intensities. Wider implications. We also discuss two possible mechanisms for the generation of breathers and negative-temperature states in experimental setups without artificial changes of the sign of the energy. These are the introduction of boundary dissipations and the free expansion of wave-packets initially in equilibrium at a positive temperature.