Hamiltonian Structure and Statistically Relevant Conserved Quantities for the Truncated Burgers-Hopf Equation

In dynamical systems with intrinsic chaos, many degrees of freedom, and many conserved quantities, a fundamental issue is the statistical relevance of suitable subsets of these conserved quantities in appropriate regimes. The Galerkin truncation of the Burgers-Hopf equation has been introduced recently as a prototype model with solutions exhibiting intrinsic stochasticity and a wide range of correlation scaling behavior that can be predicted successfully by simple scaling arguments. Here it is established that the truncated Burgers-Hopf model is a Hamiltonian system with Hamiltonian given by the integral of the third power. This additional conserved quantity, beyond the energy, has been ignored in previous statistical mechanics studies of this equation. Thus, the question arises of the statistical significance of the Hamiltonian beyond that of the energy. First, an appropriate statistical theory is developed that includes both the energy and Hamiltonian. Then a convergent Monte Carlo algorithm is developed for computing equilibrium statistical distributions. The probability distribution of the Hamiltonian on a microcanonical energy surface is studied through the Monte-Carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the Hamiltonian. Empirical numerical estimates and simple analysis are combined to demonstrate that the statistically relevant values of the Hamiltonian have vanishingly small measure as the number of degrees of freedom increases with fixed mean energy. The predictions of the theory for relevant and irrelevant values for the Hamiltonian are confirmed through systematic numerical simulations. For statistically relevant values of the Hamiltonian, these simulations show a surprising spectral tilt rather than equipartition of energy. This spectral tilt is predicted and confirmed independently by Monte Carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt. On the other hand, the theoretically predicted correlation scaling law is satisfied both for statistically relevant and irrelevant values of the Hamiltonian with excellent accuracy. The results established here for the Burgers-Hopf model are a prototype for similar issues with significant practical importance in much more complex geophysical applications. Several interesting mathematical problems suggested by this study are mentioned in the final section.