## Abstract

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 which 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 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 which 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.

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This work was partly supported by
the National Science Foundation through grants DMS-9502142 and
DMS-9972865-01. Gregor Kovacic would like to thank the faculty ans staff
of the Courant Institute for their hospitality during the academic year 2000/2001.

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