  Classical nonlinear waves exhibit a phenomenon of condensation that results
from the natural irreversible process of thermalization, in analogy with the
quantum Bose-Einstein condensation. Wave condensation originates in the
divergence of the thermodynamic equilibrium Rayleigh-Jeans distribution, which
is responsible for the macroscopic population of the fundamental mode of the
system. However, achieving complete thermalization and condensation of
incoherent waves through nonlinear optical propagation is known to require
prohibitive large interaction lengths. Here, we derive a discrete kinetic
equation describing the nonequilibrium evolution of the random wave in the
presence of a structural disorder of the medium. Our theory reveals that a weak
disorder accelerates the rate of thermalization and condensation by several
order of magnitudes. Such a counterintuitive dramatic acceleration of
condensation can provide a natural explanation for the recently discovered
phenomenon of optical beam self-cleaning. Our experiments in multimode optical
fibers report the observation of the transition from an incoherent thermal
distribution to wave condensation, with a condensate fraction of up to 60% in
the fundamental mode of the waveguide trapping potential.
