Richard Kleeman, Bruce E. Turkington
Exact spectral truncations of the unforced, inviscid Burgers-Hopf equation are Hamiltonian systems with many degrees of freedom that exhibit instrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to validate stochastic mode reduction strategies. In the present work the Burger-Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed set of evolution equations for the mean values of the set of m << n lowest modes is derived. From the perspective of nonequilibrium statistical mechanics this model reduction is a coarse-graining from n-mode microstates to m-mode macrostates. The nonequilibrium model is constructed by associating paths of trial probability densities on phase space with evolving macrostates, and fitting these paths to the underlying microscopic dynamics by minimizing a time-integrated, mean-squared norm of their residual with respect to the Liouville equation. For the truncated Burgers-Hopf equation, the best-fit macrodynamics has fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined by the statistical closure up to a single adjustable parameter. The accuracy and range of validity of this closure is assessed by direct numerical simulations of statistical ensembles of microscopic solutions, and the predicted behaviour is shown to be well represented by the reduced equations
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http://arxiv.org/abs/1206.6545
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