Statistical mechanics for truncations of the Burgers-Hopf equation: A model for intrinsic stochastic behavior with scaling

In this paper we consider both analytically and numerically several finite-dimensional approximations for the inviscid Burgers-Hopf equation. Fourier Galerkin truncation is introduced and studied as a simple one-dimensional model with intrinsic chaos and a well-defined mathematical structure allowing for an equilibrium statistical mechanics formalism. A simple scaling theory for correlations is developed that is supported strongly by the numerical evidence. Several semi-discrete difference schemes with similar mathematical properties conserving discrete momentum and energy are also considered. The mathematical properties of the difference schemes are analyzed and the behavior of the difference schemes is compared and contrasted with the Fourier Galerkin truncation. Numerical simulations are presented which show similarities and subtle differences between different finite-dimensional approximations both in the deterministic and stochastic regimes with many degrees of freedom.