On Normal and Non-Normal Wave Statistics Implied by a Canonical–Microcanonical Gibbs Ensemble of the Truncated KdV System

The truncated Korteweg–De Vries (TKdV) system—a shallow-water wave model with Hamiltonian structure that exhibits weakly turbulent dynamics—has been found to accurately predict the anomalous wave statistics observed in recent laboratory experiments. Majda et al. (Proc Natl Acad Sci 116(10):3982–3987, 2019) developed a TKdV statistical mechanics framework based on a mixed Gibbs measure that is supported on a surface of fixed energy (microcanonical) and takes the usual canonical form in the Hamiltonian. This paper reports two rigorous results regarding the surface-displacement distributions implied by this ensemble, both in the limit of the cutoff wavenumber Λ growing large. First, we prove that if the inverse temperature vanishes, displacement statistics converge to Gaussian as Λ → ∞ . Second, we prove that if nonlinearity is absent and the inverse-temperature satisfies a certain physically-motivated scaling law, then displacement statistics converge to Gaussian as Λ → ∞ . When the scaling law is not satisfied, simple numerical examples demonstrate symmetric, yet highly non-Gaussian, displacement statistics to emerge in the linear system, illustrating that nonlinearity is not a strict requirement for non-normality in the fixed-energy ensemble. The new results, taken together, imply necessary conditions for the anomalous wave statistics observed in previous numerical studies. In particular, non-vanishing inverse temperature and either the presence of nonlinearity or the violation of the scaling law are required for displacement statistics to deviate from Gaussian. The proof of this second theorem involves the construction of an approximating measure, which we find also elucidates the peculiar spectral decay observed in numerical studies and may open the door for improved sampling algorithms.