Hyperskewness of (1  +  1)-dimensional KPZ height fluctuations

Singha, Tapas; Nandy, Malay K.

doi:10.1088/1742-5468/2016/01/013203

Cited by 5 publications

(7 citation statements)

References 53 publications

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“…The Gaussian behavior obtained for Eq. ( 3) coincides with analytical DRG results which follow successful analyses of field statistics for the KPZ [45][46][47] and nonlinear-Molecular Beam Epitaxy (MBE) [48] equations, and for the scalar Burgers equation with non-conserved noise [49]. The method performs a partial RG transformation in which the equation is coarse-grained [50], while omitting the rescaling step [27,36].…”

supporting

confidence: 74%

The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class

Rodriguez-Fernandez,

Cuerno

2019

Preprint

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The Kardar-Parisi-Zhang (KPZ) equation is a paradigmatic model of nonequilibrium low-dimensional systems with spatiotemporal scale invariance, recently highlighting universal behavior in fluctuation statistics. Its space derivative, namely the noisy Burgers equation, has played a very important role in its study, predating the formulation of the KPZ equation proper, and being frequently held as an equivalent system. We show that, while differences in the scaling exponents for the two equations are indeed due to a mere space derivative, the field statistics behave in a remarkably different way: while KPZ follows the Tracy-Widom distribution, its derivative displays Gaussian behavior, hence being in a different universality class. We reach this conclusion via direct numerical simulations of the equations, supported by a dynamic renormalization group study of field statistics.

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supporting

confidence: 74%

The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class

Rodriguez-Fernandez,

Cuerno

2019

Preprint

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“…This scheme of calculation finds out the renormalized quantities directly from various loop diagrams for the second and third order cumulants of velocity derivative. This type of scheme has previously been used for the calculation of statistical cumulants in KPZ [35,36,37] and VLDS [38] surface growth dynamics. Employing diagrammatic approach, we have seen that there are two contributing Feynman diagrams [ Fig.…”

Section: Discussionmentioning

confidence: 99%

“…This scheme is different from the above RG schemes as it finds a relation between the renormalized Feynman diagrams involving the renormalized viscosity. Recently this procedure was found to be successful in determining the experimentally observed statistical characteristics in Kardar-Parisi-Zhang (KPZ) [35,36,37] and Villain-Lai-Das Sarma (VLDS) [38] surface growth dynamics. This scheme enables us to calculate the second-and third-order cumulants of velocity derivative and the resulting value for skewness is obtained as S = −0.647.…”

Section: Introductionmentioning

confidence: 99%

Renormalized cumulants and velocity derivative skewness in Kolmogorov turbulence

Singha¹,

Dutta²,

Nandy³

2017

J. Phys. A: Math. Theor.

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We apply a renormalized perturbative scheme on the Navier-Stokes equation for an incompressible isotropic turbulent velocity field. This allows us to obtain the renormalized expressions for second-and third-order cumulants of the velocity derivative directly from the corresponding Feynman diagrams. The resulting expressions are integrated numerically by excluding and including the dissipation range assuming Kolmogorov and Pao's phenomenological expressions for the energy spectrum. The ensuing values for skewness are found to be S = −0.647 (when the dissipation range is excluded) and S = −0.682 (when the dissipation is included). These estimated values are compared with various experimental, numerical, and theoretical results.

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“…where k n = − n−1 j=1 k j , ω n = − n−1 j=1 ω j . The function L n is perturbatively computed to one loop order [43][44][45][46] as L n = (2D)δ n,2 + L n,1 ,…”

Section: Cumulantsmentioning

confidence: 99%

“…While λ and D do not renormalize and are thus scaleindependent, the coarse-grained linear coefficient ν(k) depends on wave vector as ν(k) ≃ (6Dλ 2 /π) 1/3 |k| −1 (see Appendix A 1 for details). We exploit this fact to estimate by DRG the cumulants of the statistical distribution of φ, following the methodology successfully employed for the KPZ [43][44][45] and NLMBE [46] equations. Thus, the n-th cumulant reads…”

Section: A Drg Evaluation Of Cumulantsmentioning

confidence: 99%

Gaussian statistics as an emergent symmetry of the stochastic scalar Burgers equation

Rodríguez-Fernández

Cuerno

2019

Phys. Rev. E

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Symmetries play a conspicuous role in the large-scale behavior of critical systems. In equilibrium they allow to classify asymptotics into different universality classes and, out of equilibrium, they sometimes emerge as collective properties which are not explicit in the "bare" interactions. Here we elucidate the emergence of an up-down symmetry in the asymptotic behavior of the stochastic scalar Burgers equation in one and two dimensions, manifested by the occurrence of Gaussian fluctuations even within the time regime controlled by nonlinearities. This robustness of Gaussian behavior contradicts naive expectations, due to the detailed relation -including the lack of up-down symmetry-between Burgers equation and the Kardar-Parisi-Zhang equation, which paradigmatically displays non-Gaussian fluctuations described by Tracy-Widom distributions. We reach our conclusions via a dynamic renormalization group study of the field statistics, confirmed by direct evaluation of the field probability distribution function from numerical simulations of the dynamical equation.

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Hyperskewness of (1 + 1)-dimensional KPZ height fluctuations

Cited by 5 publications

References 53 publications

The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class

The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class

Renormalized cumulants and velocity derivative skewness in Kolmogorov turbulence

Gaussian statistics as an emergent symmetry of the stochastic scalar Burgers equation

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