  When a periodic 1D system described by a tight-binding model is uniformly
initialized with equal amplitudes at all sites, yet with completely random
phases, it evolves into a thermal distribution with no spatial correlations.
However, when the system is nonlinear, correlations are spontaneously formed.
We find that for strong nonlinearities, the intensity histograms approach a
narrow Gaussian distributed around their mean and phase correlations are formed
between neighboring sites. Sites tend to be out-of-phase for a positive
nonlinearity and in-phase for a negative one. The field correlations take a
universal shape independent of parameters. This nonlinear evolution produces an
effectively dynamically disordered potential which exhibits interesting
diffusive behavior.
