Wavelet imaging of transient energy localization in nonlinear systems at thermal equilibrium: the case study of NaI crystals at high temperature

While intrinsic localized modes (ILM) have been predicted to exist in many systems at zero temperature, detecting and quantifying their transient excitation in crystals at thermal equilibrium is an elusive problem and a different task altogether. While some indirect evidence of spontaneous excitation of ILMs has been collected by molecular dynamics (MD) simulations of alkali halides crystals at thermal equilibrium, more stringent numerical evidence and undisputed experimental proof are still lacking. In this paper we introduce a method to resolve transient excitations in time-frequency space. Our technique is based on continuous wavelet transform of velocity time series coupled to a threshold-dependent filtering procedure to isolate excitation events from background noise in a given spectral region. By following in time the center of mass of the reference frequency interval, the data can be easily exploited to investigate the statistics of burst excitation, by computing, for instance, the distribution of the lifetimes, excitation times, amplitudes and energies. As an illustration of our method, we investigate transient excitations in the gap of NaI crystals at thermal equilibrium at different temperatures. Our results reveal complex ensembles of transient nonlinear bursts in the gap, whose lifetime and excitation rate increase with temperature. Furthermore, a comparison with previous theoretical results on zero-temperature modes seems to lend support to the claim that a small subset of the recorded bursts might be ILMs polarized along the [111] direction. All in all, our method is a powerful tool to investigate transient excitations in many-body systems at thermal equilibrium. Our procedure gives access to both the equilibrium and the kinetics of the transient excitation process, allowing one in principle to reconstruct the full dynamical picture.