Statistical mechanics for truncations of the Burgers-Hopf equation

A model for intrinsic stochastic behavior with scaling

A. Majda, I. Timofeyev

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)39-96
Number of pages58
JournalMilan Journal of Mathematics
Volume70
Issue number1
StatePublished - 2002

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Difference Scheme
Statistical Mechanics
Truncation
Finite-dimensional Approximation
Scaling
Galerkin
Scaling Theory
One-dimensional Model
Well-defined
Chaos
Momentum
Degree of freedom
Model
Numerical Simulation
Energy

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Statistical mechanics for truncations of the Burgers-Hopf equation : A model for intrinsic stochastic behavior with scaling. / Majda, A.; Timofeyev, I.

In: Milan Journal of Mathematics, Vol. 70, No. 1, 2002, p. 39-96.

Research output: Contribution to journalArticle

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