Statistical mechanics for truncations of the Burgers-Hopf equation: a model for intrinsic stochastic behavior with scaling
type: Item
instance of: scholarly article
title: Statistical mechanics for truncations of the Burgers-Hopf equation: a model for intrinsic stochastic behavior with scaling
zbMATH Open document ID: 1049.35017
author: Andrew J. Majda
author: Ilya Timofeyev
published in: Milan Journal of Mathematics
publication date: 2004-09-07T00:00:00Z
review text: Models for one-dimensional dynamics approximating the Hopf, or inviscid Burgers, equation \(u_t + \frac{1}{2}(u^2)_x = 0\) are investigated both analytically and numerically. Continuing their paper [Proc. Natl. Acad. Sci. USA 97, No. 23, 12413--12417 (2000; Zbl 0969.35117)] the authors study the Fourier Galerkin truncated spectral approximations resulting in systems of nonlinear ODE's allowing for an equilibrium statistical mechanics formulation. Computer simulation, in several parameter regimes and with random initial data, confirms the statistical predictions and scaling behaviour of correlation times. Other discrete approximations involving spatially discrete finite-difference schemes are analyzed as well. A new simple 5-points scheme is introduced and compared with those of \textit{J. Goodman} and \textit{P. D. Lax} [Commun. Pure Appl. Math. 41, No. 5, 591--613 (1988; Zbl 0647.65062)] and \textit{N. J. Zabusky} and \textit{M. D. Kruskal} [Phys. Rev. Lett. 15, 240--243 (1965)]. The new one and the latter one have intrinsic stochastic dynamics with numerical evidence for ergodicity and mixing. These methods and the Fourier-Galerkin truncation determine different nonlinear dynamics at lower wave numbers. The models demostrate that predictions can be made by simple mathematical arguments and scaling theory due to longer correlation times on the larger scales. This fits the pattern of stochastic modelling of complex systems by reduced systems involving only the degrees of freedom with longer correlation time. The style of the paper avoids proving theorems, it is a discussion of research results mingling mathematical theory and scientific computing.
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 37N10
Mathematics Subject Classification ID: 35F20
Mathematics Subject Classification ID: 65M60
zbMATH DE Number: 2099228
zbMATH Keywords: Fourier-Galerkin truncated spectral approximation
zbMATH Keywords: scaling theory for correlations
zbMATH Keywords: difference schemes
zbMATH Keywords: numerical simulations
zbMATH Keywords: intrinsic stochastic dynamics
zbMATH Keywords: ergodicity and mixing
reviewed by: Alexander V. Bulinski
MaRDI profile type: Publication
full work available at URL: s00032-002-0003-9
OpenAlex ID: W2143079935
DOI: 10.1007/S00032-002-0003-9
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description: scientific article; zbMATH DE number 2099228