Premium
A Nonequilibrium Statistical Model of Spectrally Truncated Burgers‐Hopf Dynamics
Author(s) -
Kleeman Richard,
Turkington Bruce E.
Publication year - 2014
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21498
Subject(s) - mathematics , burgers' equation , statistical physics , nonlinear system , closure (psychology) , hamiltonian (control theory) , scaling , hopf bifurcation , degrees of freedom (physics and chemistry) , reduction (mathematics) , phase space , mathematical analysis , partial differential equation , physics , mathematical optimization , quantum mechanics , geometry , bifurcation , economics , market economy
Exact spectral truncations of the unforced, inviscid Burgers‐Hopf equation are Hamiltonian systems with many degrees of freedom that exhibit intrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies. In the present paper the Burgers‐Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed system of evolution equations for the mean values of the m lowest modes is derived for m ≪ n . In the reduced model the m ‐mode macrostates are associated with trial probability densities on the phase space of the n ‐mode microstates, and a cost functional is introduced to quantify the lack of fit of paths of these densities to the Liouville equation. The best‐fit macrodynamics is obtained by minimizing the cost functional over paths, and the equations governing the closure are then derived from Hamilton‐Jacobi theory. The resulting reduced equations have a fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined up to a single closure parameter. The accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles, and the predicted behavior is found to be well represented by the reduced equations. © 2014 Wiley Periodicals, Inc.