  We propose and demonstrate, with the one-dimensional Korteweg-de
Vries-Burgers model, the scenarios of transfer loop and \textit{all-scale}
statistical equilibrium, the former being associated to shock formation and the
latter to Gaussian distributions as in a canonical ensemble, but with
wavelength-dependent temperatures. The discussions emphasize, among the
multi-disciplinary relevance, the classical nonlinear acoustics and quantum
shock waves, for the possibility of more favorable experimental tests.
