Experimental determination of KPZ height-fluctuation distributions

Miettinen, Lasse; Myllys, Markko; Merikoski, J.; Timonen, J.

doi:10.1140/epjb/e2005-00235-y

Cited by 72 publications

(101 citation statements)

References 17 publications

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“…Takeuchi [42] obtained, via a numerical simulation of the PNG model and an experiment on the TLC, a universal function that undergoes a crossover from a transient state to the stationary regime. Miettinen et al [23] carried out an experimental study of the slow propagation of the combustion front on a sheet of paper to investigate upon the front fluctuation distribution. Their experimental data for the transient and stationary states were found to fit well with the GOE-TW and F 0 distributions, respectively.…”

Section: Discussionmentioning

confidence: 99%

Kurtosis of height fluctuations in (1 + 1) dimensional KPZ dynamics

Singha¹,

Nandy²

2015

J. Stat. Mech.

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Section: Discussionmentioning

confidence: 99%

Kurtosis of height fluctuations in (1 + 1) dimensional KPZ dynamics

Singha¹,

Nandy²

2015

J. Stat. Mech.

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“…A separate, critical paper by Prähofer and Spohn [57] introduced the notion of the Airy process, bringing spatial correlations into the RMT context, and tightening ties between KPZ physics and TW mathematical communities. During this period, flameless Finnish firefronts continued to burn [58,59,60], mode-coupling [61] and nonperturbative [62,63] approaches developed further, and a suggestive Dutch chemical vapor deposition experiment appeared [64], providing evidence of an asymmetric 2+1 KPZ height fluctuation PDF. For a recent example of the nice interplay of KPZ/DPRM statistical physics and TW/BR mathematics, as well as a brief account of the foundational Ulam-LIS numerics of Baer and Brock, see [65]; note, too- [66,67].…”

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confidence: 98%

A KPZ Cocktail-Shaken, not Stirred...

2015

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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):

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“…The KPZ class has been explored in several recent experiments [4,5], and the DP has found applications ranging from biophysics [6] to describing the glass phase of pinned vortex lines [7] and magnetic walls [8]. The height of the growing interface, h(x, t), corresponds to the free energy of a DP of length t starting at point x, under a mapping which is exact in the continuum (Cole-Hopf), as well as for some discrete realizations.…”

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Directed polymer near a hard wall and KPZ equation in the half-space

Gueudré¹,

Doussal²

2012

EPL

131

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Abstract. -We study the directed polymer with fixed endpoints near an absorbing wall, in the continuum and in presence of disorder, equivalent to the KPZ equation on the half space with droplet initial conditions. From a Bethe Ansatz solution of the equivalent attractive boson model we obtain the exact expression for the free energy distribution at all times. It converges at large time to the Tracy Widom distribution F4 of the Gaussian Symplectic Ensemble (GSE). We compare our results with numerical simulations of the lattice directed polymer, both at zero and high temperature.Much progress was achieved recently in finding exact solutions in one dimension for noisy growth models in the Kardar-Parisi-Zhang (KPZ) universality class [1,2], and for the closely related equilibrium statistical mechanics problem of the directed polymer (DP) in presence of quenched disorder [3]. The KPZ class has been explored in several recent experiments [4,5], and the DP has found applications ranging from biophysics [6] to describing the glass phase of pinned vortex lines [7] and magnetic walls [8]. The height of the growing interface, h(x, t), corresponds to the free energy of a DP of length t starting at point x, under a mapping which is exact in the continuum (Cole-Hopf), as well as for some discrete realizations. Not only the scaling exponents h ∼ t 1/3 , x ∼ t 2/3 are known [9,10], but also the one-point (and in some cases the manypoint) probability distribution (PDF) of the height have been obtained [16,17]. Their dependence in the initial condition was found to exhibit remarkable universality at large time, with only a few subclasses, most being related to Tracy Widom (TW) distributions [15] of largest eigenvalues of random matrices. Most of these subclasses were initially discovered in a discrete growth model (the PNG model) [11][12][13] which can be mapped onto the statistics of random permutations [14], and a zero temperature lattice DP model [10]. Recently, exact solutions have been obtained directly in the continuum at arbitrary time t, for the droplet [18][19][20][21], flat [22,23] and stationary [24] initial conditions. The PDF of the height h(x, t) converges at large time to F 2 , the Gaussian unitary ensemble (GUE), and to F 1 , the Gaussian orthogonal ensemble (GOE) universal TW distributions, for droplet and flat initial conditions respectively. One useful method which led to these solutions introduces n replica and maps the DP problem to the Lieb Liniger model, i.e. the quantum mechanics of n bosons with mutual delta-function attraction, a model which can be solved using the Bethe Ansatz.The KPZ equation on the half line x > 0, equivalently a DP in presence of a wall, is also of great interest. In the statistical mechanics context constrained fluctuations are important for the study of fluctuation-induced (Casimir) forces [25,26] and for extreme value statistics. In the surface growth context one can study an interface pinned at a point, or an average growth rate which jumps across a boundary. The half space probl...

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Experimental determination of KPZ height-fluctuation distributions

Cited by 72 publications

References 17 publications

Kurtosis of height fluctuations in (1 + 1) dimensional KPZ dynamics

Kurtosis of height fluctuations in (1 + 1) dimensional KPZ dynamics

A KPZ Cocktail-Shaken, not Stirred...

Directed polymer near a hard wall and KPZ equation in the half-space

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