Critical exponents of the KPZ equation via multi-surface coding numerical simulations

Marinari, Enzo; Pagnani, Andrea; Parisi, Giorgio

doi:10.1088/0305-4470/33/46/303

Cited by 140 publications

(235 citation statements)

References 26 publications

(46 reference statements)

Supporting

Mentioning

176

Contrasting

Order By: Relevance

“…However, the above estimate also has an intersection with the best previous estimate of z for the KPZ class, in the range [1.605, 1.64] [32]. For the other values of F , the estimates of z also intercept this KPZ range.…”

Section: Numerical Study Of Roughness Scalingmentioning

confidence: 76%

“…In order to test this scaling approach, we estimated the amplitudes A and B in the limit L → ∞ by extrapolating the ratios W sat (L)/L α and τ (L)/L z , respectively. The values α = 0.385 and z = 1.615 of the KPZ class [32] were used in the calculation of those quantities. Following the notation of Ref.…”

Section: Scaling Theory For the Crossover Of Surface Roughnessmentioning

confidence: 99%

See 1 more Smart Citation

Surface and bulk properties of deposits grown with a bidisperse ballistic deposition model

Silveira

Reis

2007

Phys. Rev. E

View full text Add to dashboard Cite

We study roughness scaling of the outer surface and the internal porous structure of deposits generated with the three-dimensional bidisperse ballistic deposition (BBD), in which particles of two sizes are randomly deposited. Systematic extrapolation of roughness and dynamical exponents and the comparison of roughness distributions indicate that the top surface has Kardar-ParisiZhang scaling for any ratio F of the flux between large and small particles. A scaling theory predicts the characteristic time of the crossover from random to correlated growth in BBD and provides relations between the amplitudes of roughness scaling and F in the KPZ regime. The porosity of the deposits monotonically increases with F and scales as F 1/2 for small F , which is also explained by the scaling approach and illustrates the possibility of connecting surface growth rules and bulk properties. The suppression of relaxation mechanisms in BBD enhances the connectivity of the deposits when compared to other ballisticlike models, so that they percolate down to F ≈ 0.05.The fractal dimension of the internal surface of the percolating deposits is D F ≈ 2.9, which is very close to the values in other ballisticlike models and suggests universality among these systems.

show abstract

Section: Numerical Study Of Roughness Scalingmentioning

confidence: 76%

Section: Scaling Theory For the Crossover Of Surface Roughnessmentioning

confidence: 99%

Surface and bulk properties of deposits grown with a bidisperse ballistic deposition model

Silveira

Reis

2007

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…One of the main results of our analysis indicates that the numerical solution obtained with a non-Galilean-invariant finite-difference scheme yields the same critical exponents as the well known ones obtained with the standard discretization scheme [Eq. (15)]. Moreover, even the discretization used in [?, ?…”

Section: It Is a Trivial Task To Verify That The Laplacian Is (∂mentioning

confidence: 99%

KPZ equation: Galilean-invariance violation, consistency, and fluctuation-dissipation issues in real-space discretization

Wio

Revelli²,

Deza³

et al. 2010

Europhys. Lett.

View full text Add to dashboard Cite

Abstract. -Strong constraints are drawn for the choice of real-space discretization schemes, using the known fact that the KPZ equation results from a diffusion equation (with multiplicative noise) through a Hopf-Cole transformation. Whereas the nearest-neighbor discretization passes the consistency tests, known examples in the literature do not. We emphasize the importance of the Lyapunov functional as natural starting point for real-space discretization and, in the light of these findings, challenge the mainstream opinion on the relevance of Galilean invariance.The KPZ equation is nowadays a paradigm as a stochastic-field description of a vast class of nonequilibrium phenomena, largely transcending the realm of surface growth processes in which it was originally formulated [?, ?, ?]. This stochastic nonlinear partial differential equation governs the evolution of a field h (x, t), that describes the height of a fluctuating interface. For a one-dimensional (1D) homogeneous substrate of size L, the KPZ equation readsξ(x, t) being a Gaussian white noise with ξ(x, t) = 0 and ξ(x, t)ξ(x , t ) = 2δ(x − x )δ(t − t ), F is an external constant driving force, ν determines the surface tension, and λ is proportional to the average growth velocity. As usual, periodic boundary conditions (pbc) are assumed. There are two main symmetries associated with the 1D KPZ equation: Galilean invariance and the fluctuation-dissipation relation. On one hand, Galilean invariance has been traditionally linked to the exactness of the relation α + z = 2 among the critical exponents, in any spatial dimensionality [the roughness exponent α, characterizing the surface morphology in the stationary regime, and the dynamic exponent z, indicating the correlation p-1

show abstract

“…Complementary experimental work was scant, but provocative-see early, comprehensive reviews [28,29,30]. This epoch closed with the beginnings of Finnish investigations [31] of kinetically-roughened KPZ firelines, key mathematical papers [32,33], as well as refreshing nonperturbative [34] and conformally invariant [35] perspectives, the former inspiring numerical rebuttals [36,37,38,39] of a stubborn, battered suggestion [40], revived nevertheless shortly thereafter [41], that 4+1 might be the upper critical dimension (UCD) of the KPZ problem. Recent numerics [42,43,44,45] appear to have buried this idea, but from the analytical side [46,47], there's lingering suspicion that something nontrivial is afoot near this particular dimension.…”

mentioning

confidence: 99%

A KPZ Cocktail-Shaken, not Stirred...

2015

View full text Add to dashboard Cite

The stochastic partial differential equation proposed nearly three decades ago by Kardar, Parisi and Zhang (KPZ) continues to inspire, intrigue and confound its many admirers. Here, we i) pay debts to heroic predecessors, ii) highlight additional, experimentally relevant aspects of the recently solved 1+1 KPZ problem, iii) use an expanding substrates formalism to gain access to the 3d radial KPZ equation and, lastly, iv) examining extremal paths on disordered hierarchical lattices, set our gaze upon the fate of d=∞ KPZ. Clearly, there remains ample unexplored territory within the realm of KPZ and, for the hearty, much work to be done, especially in higher dimensions, where numerical and renormalization group methods are providing a deeper understanding of this iconic equation.Keywords Nonequilibrium Growth · Extremal Paths · Universal Limit Distributions In a NutshellThe history of physics has been punctuated at seminal moments by the appearance of certain fundamental equations (and associated models) which have vigorously propelled the enterprise forward, serving as an explosively rich departure point, generating a myriad of alternative perspectives, creative insights, surprising connections and, given its sustained impregnability, often remain for many years a sacred object of fascination and obsession to its dedicated disciples. Recent, obvious suspects in this regard include the quantum mechanical Schrödinger equation (equally well, its flipside-Feynman's path integral formulation), or the wonderfully elusive Navier-Stokes equation governing fluid mechanics, by which may be gleaned the scaling secrets of turbulent flow, dynamically encoded in the whirling eddies drawn by da Vinci centuries ago. Within the domain of equilibrium statistical physics, the 2d Ising Model, with its tour-de-force algebraic solution by Onsager, followed by combinatoric, graphical, Grassmannian, Monte Carlo, as well as full-blown field-theoretic, scaling, and renormalization group treatments, represents an extraordinary legacy that continues unabated to this very day. Arguably, a non-equilibrium statistical mechanical analogue to Ising/Onsager is the iconic equation [1] proposed a generation ago by Kardar, Parisi, and Zhang (KPZ); it captures the statistical fluctuations of a kinetically-roughened scalar height field h(x, t):

show abstract

Critical exponents of the KPZ equation via multi-surface coding numerical simulations

Cited by 140 publications

References 26 publications

Surface and bulk properties of deposits grown with a bidisperse ballistic deposition model

Surface and bulk properties of deposits grown with a bidisperse ballistic deposition model

KPZ equation: Galilean-invariance violation, consistency, and fluctuation-dissipation issues in real-space discretization

A KPZ Cocktail-Shaken, not Stirred...

Contact Info

Product

Resources

About