Remarkable statistical behavior for truncated Burgers–Hopf dynamics

Majda, Andrew J.; Timofeyev, Ilya

doi:10.1073/pnas.230433997

Cited by 92 publications

(162 citation statements)

References 17 publications

(12 reference statements)

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“…Several illuminating exact solutions of the Galerkin approximation to (1.1) also are discussed in that section. The material in Section 3 is an expanded version of the discussion in the authors' very recent paper [19]. Section 4 contains numerical evidence for ergodicity and correlation scaling for Galerkin truncations which strongly confirms the predictions of the theory in Section 3.…”

supporting

confidence: 57%

Statistical Mechanics for Truncations of the Burgers-Hopf Equation: A Model for Intrinsic Stochastic Behavior with Scaling

Majda¹,

Timofeyev

2002

Milan Journal of Mathematics

103

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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.

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supporting

confidence: 57%

Statistical Mechanics for Truncations of the Burgers-Hopf Equation: A Model for Intrinsic Stochastic Behavior with Scaling

Majda¹,

Timofeyev

2002

Milan Journal of Mathematics

103

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“…It was noted in [15] that there are invariant subspaces for this system. Observing (42) figure 4 does not.…”

Section: Invariant Subspacesmentioning

confidence: 88%

“…This is not the case because when one does a Fourier-Galerkin truncation the theory becomes non-integrable and incredibly rich [15][16][17]. The next step is to express the fluid's equation in terms of x and τ by using the partial derivative relation…”

Section: Relativistic Ideal Fluid Mechanics and The Inviscid Burgers mentioning

confidence: 99%

“…The Galerkin truncated version of this equation has been shown to have a number of interesting properties in numerical experiments [15][16][17]33] including ergodic chaotic behavior mimicking turbulence (see table 1). …”

Section: Background On the Burgers Equationmentioning

confidence: 99%

“…In reviewing it extensively, this author was drawn to recent research on the inviscid Burgers equation, and in particular to some very interesting numerical results of simulations to this equation using Galerkin truncation [15][16][17]. This paper is mainly about extending these results and asking if they might be indications of an emerging quantum behavior.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The quark-gluon plasma and the stochastic interpretation of quantum mechanics

Davidson¹

2010

Physica E: Low-dimensional Systems and Nanostructures

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Quark-gluon plasmas formed in heavy ion collisions at high energies are well described by ideal classical fluid equations with nearly zero viscosity. It is believed that a similar fluid permeated the entire universe at about three microseconds after the big bang. The estimated Reynolds number for this quark-gluon plasma at 3 microseconds is approximately 10^19. The possibility that quantum mechanics may be an emergent property of a turbulent proto-fluid is tentatively explored. A simple relativistic fluid equation which is consistent with general relativity and is based on a cosmic dust model is studied. A proper time transformation transforms it into an inviscid Burgers equation. This is analyzed numerically using a spectral method. Soliton-like solutions are demonstrated for this system, and these interact with the known ergodic behavior of the fluid to yield a stochastic and chaotic system which is time reversible. Various similarities to quantum mechanics are explored.

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A Nonequilibrium Statistical Model of Spectrally Truncated Burgers‐Hopf Dynamics

Kleeman

Turkington

2013

Comm Pure Appl Math

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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.

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Remarkable statistical behavior for truncated Burgers–Hopf dynamics

Cited by 92 publications

References 17 publications

Statistical Mechanics for Truncations of the Burgers-Hopf Equation: A Model for Intrinsic Stochastic Behavior with Scaling

Statistical Mechanics for Truncations of the Burgers-Hopf Equation: A Model for Intrinsic Stochastic Behavior with Scaling

The quark-gluon plasma and the stochastic interpretation of quantum mechanics

A Nonequilibrium Statistical Model of Spectrally Truncated Burgers‐Hopf Dynamics

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