Invariant measures and entropy production in wave turbulence

Jakobsen, Per Kristen; Newell, Alan C.

doi:10.1088/1742-5468/2004/10/l10002

Cited by 16 publications

(42 citation statements)

References 9 publications

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“…Can one show that a natural closure occurs in L d or, if not, how the natural closure arises in taking the L → ∞ limit? (Jakobsen and Newell, 2004).…”

Section: Continuum Limit Of Finite Dimensional Wave Turbulencementioning

confidence: 99%

Wave Turbulence: A Story Far from Over

Newell¹,

Rumpf²

2013

World Scientific Series on Nonlinear Science Series A

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The goals of this chapter are: To state and review the premises on which a successful asymptotic closure of the BBGKY hierarchy of moment/ cumulant equations is based; to describe how and why this closure is attained; to examine the nature of solutions of the kinetic equation; to discuss obstacles which limit the theory's validity and suggest how the theory might then be modified; to compare the experimental evidence in a range of applications with the theory's predictions; and finally, and most importantly, to suggest open challenges and to encourage the reader to apply and explore wave turbulence with confidence.

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“…Can one show that a natural closure occurs in L d or, if not, how the natural closure arises in taking the L → ∞ limit? (Jakobsen and Newell, 2004).…”

Section: Continuum Limit Of Finite Dimensional Wave Turbulencementioning

confidence: 99%

Wave Turbulence: A Story Far from Over

Newell¹,

Rumpf²

2013

World Scientific Series on Nonlinear Science Series A

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“…intermittency [8,[25][26][27][28][29]. This seems to be the case when a more general theoretical framework [30][31][32][33][34] is required, because the nonlinearities are not small [35,36].In this communication, the complete wave-turbulence theory is developed for a fully general 4-wave system, whose hamiltonian is expressed by the following canoni- …”

mentioning

confidence: 99%

Wave-turbulence theory of four-wave nonlinear interactions

et al. 2017

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The Sagdeev-Zaslavski (SZ) equation for wave turbulence is analytically derived, both in terms of generating function and of multi-point pdf, for weakly interacting waves with initial random phases. When also initial amplitudes are random, the one-point pdf equation is derived. Such analytical calculations remarkably agree with results obtained in totally different fashions. Numerical investigations of the two-dimensional nonlinear Schrödinger equation (NLSE) and of a vibrating plate prove that: (i) generic Hamiltonian 4-wave systems rapidly attain a random distribution of phases independently of the slower dynamics of the amplitudes, vindicating the hypothesis of initially random phases; (ii) relaxation of the Fourier amplitudes to the predicted stationary distribution (exponential) happens on a faster timescale than relaxation of the spectrum (Rayleigh-Jeans distribution); (iii) the pdf equation correctly describes dynamics under different forcings: the NLSE has an exponential pdf corresponding to a quasi-gaussian solution, like the vibrating plates, that also show some intermittency at very strong forcings.Introduction Dispersive waves are ubiquitous in nature, and their nonlinear interactions make them intriguing and challenging [1,2]. Wave Turbulence is the theory that describes the statistical properties of large numbers of incoherent interacting waves, with tools such as the wave kinetic equation analytically derived in the late sixties. This equation describes the evolution of the wave spectrum in time, when homogeneity and weak nonlinearity are assumed [3][4][5]. It has been applied to numerous phenomena, including ocean waves [6][7][8], capillary waves [9, 10] Alfvén waves [11], optical waves [12] and solid oscillations [13][14][15][16][17][18]. It is the analogue of the Boltzmann equation for classical particles and it allows the RayleighJeans equilibrium state as well as non-equilibrium solutions, in terms of Kolmogorov-Zakharov (KZ) spectra [19].To characterise the invariant measure of the dynamics, that is to find the complete statistical description concerning all quantities of interest, an important step has been taken by Sagdeev and Zaslavski [20], who obtained the Brout-Prigogine equation for the probability density function (pdf) of wave turbulence [21]. More recently, this statistical framework has been nicely revisited using the diagrammatic technique [4] and performing analytical calculations, in the 3-wave case [22][23][24]. Interestingly, many experimental and theoretical results have shown that deviations from wave-turbulence predictions can be found for rare events, e.g. intermittency [8,[25][26][27][28][29]. This seems to be the case when a more general theoretical framework [30][31][32][33][34] is required, because the nonlinearities are not small [35,36].In this communication, the complete wave-turbulence theory is developed for a fully general 4-wave system, whose hamiltonian is expressed by the following canoni-

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“…The canonical momenta and coordinates are given by real and imaginary parts of A σ k = 1 √ 2 (P k +iσQ k ), and A σ k and a σ k are linked by the simple rotation in the complex plane used to obtain (13). Consistently with the general definition (26), the generating function of amplitudes and phases for finite box-size L is:…”

Section: Multimode Hierarchy Equationsmentioning

confidence: 99%

“…One can show that d d k λ(k) n L (k) converges in probability to d d k λ(k)n(k) for every bounded, continuous λ. This is sufficient to infer that the amplitude characteristic function defined in (26) satisfies…”

Section: Fields With Random Phases and Amplitudesmentioning

confidence: 99%

“…surface gravity waves or waves in vibrating elastic plates, the picture is more complicated: both numerics and experiments showed deviations from theoretical predictions, and the presence of intermittency [20,21,22,23]. This was unexpected, since WT appears as a mean-field theory, based on an initial "quasi-gaussianity", previously believed to prevent sensible deviations from gaussianity.An important step forward in this context has been the development of a more efficient formalism for non-gaussian wavefields [24,25,26, 1]. In particular, these works pointed out that probability density functions (PDF) are the relevant statistical objects to be analysed, reviving the interest in the study of PDFs in WT, that dates back to the works of Peierls, Brout, Prigogine, Zawlaski and Sagdeev [27,28,29].…”

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4-wave dynamics in kinetic wave turbulence

Chibbaro

Dematteis

Rondoni

2018

Physica D: Nonlinear Phenomena

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h i g h l i g h t s• We deal with 4-wave Hamiltonian systems in the framework of wave turbulence.• Averaging technique based on the Feynman-Wyld diagrams.• Kinetic limit : leading order equations for the statistics evolution are derived.• Random-phase and random-phase and amplitude properties preserved in time.• Powerful tool to investigate many relevant physical systems. Keywords: Weak wave turbulence Kinetic theory Intermittency a b s t r a c tA general Hamiltonian wave system with quartic resonances is considered, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. The evolution equation for the multimode characteristic function Z is obtained within an ''interaction representation'' and a perturbation expansion in the small nonlinearity parameter. A frequency renormalization is performed to remove linear terms that do not appear in the 3-wave case. Feynman-Wyld diagrams are used to average over phases, leading to a first order differential evolution equation for Z . A hierarchy of equations, analogous to the Boltzmann hierarchy for low density gases is derived, which preserves in time the property of random phases and amplitudes. This amounts to a general formalism for both the N-mode and the 1-mode PDF equations for 4-wave turbulent systems, suitable for numerical simulations and for investigating intermittency. Some of the main results which are developed here in detail have been tested numerically in a recent work.

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Invariant measures and entropy production in wave turbulence

Cited by 16 publications

References 9 publications

Wave Turbulence: A Story Far from Over

Wave Turbulence: A Story Far from Over

Wave-turbulence theory of four-wave nonlinear interactions

4-wave dynamics in kinetic wave turbulence

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