Upper Critical Dimension, Dynamic Exponent, and Scaling Functions in the Mode-Coupling Theory for the Kardar-Parisi-Zhang Equation

Abstract

We study the mode-coupling approximation for the KPZ equation in the strong coupling regime. By constructing an ansatz consistent with the asymptotic forms of the correlation and response functions we determine the upper critical dimension dc = 4, and the expansion z = 2 − (d − 4)/4 + O ((4 − d) 2 ) around dc. We find the exact z = 3/2 value in d = 1, and estimate the values z ≃ 1.62, z ≃ 1.78, in d = 2, 3. The result dc = 4 and the expansion around dc are very robust and can be derived just from a mild assumption on the relative scale on which the response and correlation functions vary as z approaches 2. where the first term represents the surface tension forces which tend to smooth the interface, the second describes the non-linear growth locally normal to the surface, and the last is a noise term intended to mimic the stochastic nature of the growth process [6]. We choose the noise to be Gaussian, with zero mean and second mo-The steady state interface profile is usually described in terms of the roughness:χ/z as x → 0, so that w grows with time like t χ/z until it saturate to L χ when t ∼ L z . χ and z are the roughness and dynamic exponent respectively.For d > 2 there are two distinct regimes, separated by a critical value of the nonlinearity coefficient. In the weak coupling regime (λ < λ c ) the non-linear term is irrelevant and the behavior is governed by the λ = 0 fixed point. The linear Edward-Wilkinson equation is recovered, for which the exponents are known exactly χ = (2 − d)/2 and z = 2. The more challenging strong coupling regime (λ > λ c ), where the non-linear term is relevant is characterised by anomalous exponents, which are the subject of this paper. From the Galilean invariance [2] (the invariance of equation (1) under an infinitesimal tilting of the surface) one can derive the relation χ + z = 2, which leaves just one independent exponent. For the special case d = 1, the existence of a fluctuation-dissipation theorem leads to the exact result χ = 1/2, z = 3/2.While we have a satisfactory understanding of the KPZ equation in d = 1 [7,8] and on the Bethe lattice [4], its behavior in the strong coupling regime when d > 1 is controversial. Such results that are known derive mainly from numerical simulations [9] of discrete models which belong to the KPZ universality class [10], and on several promising but far from conclusive analytical approaches [11][12][13][14][15][16][17][18]. Most efforts are oriented towards the determination of z as a function of d for d > 1, and to understanding whether there exists an upper critical dimension d c above which z = 2: some analytical arguments suggest the existence of a finite d c [14][15][16][17][18][19], while numerical studies [9] and real space methods [20] find no evidence of a finite upper critical dimension. It is our contention that d c = 4, and that the numerical simulations all fail to see evidence for an upper critical dimension because in high dimensions, (d ≥ 4), they have not been run for sufficiently long time for the roughness w to be i...

Nonequilibrium dynamics of a growing interface

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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].…”

Slow relaxation and aging kinetics for the driven lattice gas

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