Weak chaos in the disordered nonlinear Schrödinger chain: Destruction of Anderson localization by Arnold diffusion

► In a one-dimensional disordered chain of oscillators all normal modes are localized. ► Nonlinearity leads to chaotic dynamics. ► Chaos is concentrated on rare chaotic spots. ► Chaotic spots drive energy exchange between oscillators. ► Macroscopic transport coefficients are obtained. The subject of this study is the long-time equilibration dynamics of a strongly disordered one-dimensional chain of coupled weakly anharmonic classical oscillators. It is shown that chaos in this system has a very particular spatial structure: it can be viewed as a dilute gas of chaotic spots. Each chaotic spot corresponds to a stochastic pump which drives the Arnold diffusion of the oscillators surrounding it, thus leading to their relaxation and thermalization. The most important mechanism of equilibration at long distances is provided by random migration of the chaotic spots along the chain, which bears analogy with variable-range hopping of electrons in strongly disordered solids. The corresponding macroscopic transport equations are obtained.