Path Large Deviations for the Kinetic Theory of Weak Turbulence

Guioth, Jules; Bouchet, Freddy; Eyink, Gregory L.

doi:10.1007/s10955-022-02971-x

Cited by 6 publications

(10 citation statements)

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“…We rather adopt here a perturbative approach in ǫ from the Schrödinger equation ( 7). This approach is also classical in the wave turbulence literature [2,54,55], will appear helpful in the derivation of the dynamical large deviation theory in section 3, and has the advantage to generalize easily to the case of the kinetic theory of non-linear waves with 3-wave interactions. The first step is to express the solution of the Schrödinger equation ( 7) using an expansion in power of ǫ, such that…”

Section: Wave Kinetic Equationmentioning

confidence: 99%

“…Deriving such large deviation principles from deterministic microscopic dynamics is a fundamental endeavor in theoretical and mathematical physics. Recently, the large deviation principles for a number of classical kinetic theories, starting from first principles, have been uncovered: for discrete models that mimic dilute gases and with Boltzmann like behavior [38,39], for dilute gases related to the Boltzmann equation [1,40], for the Kac model [41,42], for plasma at length scales much smaller than the Debye length related to the Landau equation [43], for homogeneous systems with long range interactions related to the Balescu-Guernsey-Lenard equation [44], for weakly interacting waves in a homogeneous setup [2] related to the wave kinetic equation. The large deviation principles describe fluctuations but also uncover gradient structure for the deterministic kinetic equation, see [45] and a simple explanation in [1].…”

Section: Introductionmentioning

confidence: 99%

“…One aim of large deviation theory is to study rare events. In the context of wave dynamics, large deviation theory has been used to study rare events for the evolution of the empirical spectrum [2] on the kinetic time scale, but also for studying the appearance of very large amplitude waves [47], for the non-linear Schrödinger dynamics for shorter time scales. Instantons structures have been predicted and compared with experimental data taken from a 300 m long wave [47].…”

Section: Introductionmentioning

confidence: 99%

“…In section 3, we derive the path large deviation principle for the temporal variations of the local spectral density. The approach has some analogy with that of [2], with a slightly different scaling assumption, and working with the Wigner distribution to describe the wave local spectral density. We then show that this Hamiltonian satisfies the expected properties.…”

Section: Introductionmentioning

confidence: 99%

“…Its irreversibility appears as a consequence of an incomplete description, rather than as a consequence of the kinetic limit itself, or some related chaotic hypothesis. This paper follows [1] and a series of other works that derive the large deviation Hamiltonians of the main classical kinetic theories, for instance [2] for homogeneous wave kinetics. We propose here a derivation of the large deviation principle in an inhomogeneous situation, for the linear scattering of waves by a weak random potential.…”

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confidence: 96%

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Dynamical large deviations for an inhomogeneous wave kinetic theory: linear wave scattering by a random medium

Onuki,

Guioth,

Bouchet

2023

Preprint

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The wave kinetic equation predicts the averaged temporal evolution of a continuous spectral density of waves either randomly interacting or scattered by the fine structure of a medium. In a wide range of systems, the wave kinetic equation is derived from a fundamental equation of wave motion, which is symmetric through time-reversal. By contrast, the corresponding wave kinetic equations is time-irreversible: its solutions monotonically increase an entropy-like quantity. A similar paradox appears whenever one make a mesoscopic description of the evolution of a very large number of microscopic degrees of freedom, the paradigmatic example being the kinetic theory of dilute gas molecules leading to the Boltzmann equation. Since Boltzmann, it has been understood that a probabilistic understanding solves the apparent paradox. More recently, it has been understood that the kinetic description itself, at a mesoscopic level, should not break time reversal symmetry [1]. The time reversal symmetry remains a fundamental property of the mesoscopic stochastic process: without external forcing the path probabilities obey a detailed balance relation with respect to an equilibrium quasipotential. The proper theoretical or mathematical tool to derive fully this mesoscopic time reversal stochastic process is large deviation theory: a This publication is part of a special issue in homage of the memory of Krzysztof Gawȩdzki. The subject of this work is large deviation theory applied to wave turbulence. Large deviation theory applied to complex dynamics and turbulent flows was one of the subjects for which Krzysztof Gawȩdzki made a number of important contributions during the last few years, see for instance [3][4][5][6][7][8]. He taught many of us, including Freddy Bouchet, many aspects of large deviation theory. We wrote a common paper on the subject of large deviation theory and non-equilibrium quasipotentials for stochastic particles with mean field interactions [8]. Given his scientific qualities, and his deep sense of friendship, it is great pleasure for us to pay homage to Krzysztof Gawȩdzki through this modest contribution.

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Section: Wave Kinetic Equationmentioning

confidence: 99%