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We address the problem of thermalization in the presence of a time-dependentdisorder in the framework of the nonlinear Schr\"odinger (or Gross-Pitaevskii)equation with a random potential. The thermalization to the Rayleigh-Jeansdistribution is driven by the nonlinearity. On the other hand, the structuraldisorder is responsible for a relaxation toward the homogeneous equilibriumdistribution (particle equipartition), which thus inhibits thermalization(energy equipartition). On the basis of the wave turbulence theory, we derive akinetic equation that accounts for the presence of strong disorder. The theoryunveils the interplay of disorder and nonlinearity. It unexpectedly revealsthat a non-equilibrium process of condensation and thermalization can takeplace in the regime where disorder effects dominate over nonlinear effects. Wevalidate the theory by numerical simulations of the nonlinear Schr\"odingerequation and the derived kinetic equation, which are found in quantitativeagreement without using adjustable parameters. Experiments realized inmultimode optical fibers with an applied external stress evidence the processof thermalization in the presence of strong disorder.

Interplay of thermalization and strong disorder: Wave turbulence theory, numerical simulations, and experiments in multimode optical fibers