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Intermittency of Height Fluctuations and Velocity Increment of The Kardar-Parisi-Zhang and Burgers Equations with infinitesimal surface tension and Viscosity in 1+1 Dimensions

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 Publication date 2005
  fields Physics
and research's language is English




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The Kardar-Parisi-Zhang (KPZ) equation with infinitesimal surface tension, dynamically develops sharply connected valley structures within which the height derivative is not continuous. We discuss the intermittency issue in the problem of stationary state forced KPZ equation in 1+1--dimensions. It is proved that the moments of height increments $C_a = < | h (x_1) - h (x_2) |^a > $ behave as $ |x_1 -x_2|^{xi_a}$ with $xi_a = a$ for length scales $|x_1-x_2| << sigma$. The length scale $sigma$ is the characteristic length of the forcing term. We have checked the analytical results by direct numerical simulation.



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The joint probability distribution function (PDF) of the height and its gradients is derived for a zero tension $d+1$-dimensional Kardar-Parisi-Zhang (KPZ) equation. It is proved that the height`s PDF of zero tension KPZ equation shows lack of positivity after a finite time $t_{c}$. The properties of zero tension KPZ equation and its differences with the case that it possess an infinitesimal surface tension is discussed. Also potential relation between the time scale $t_{c}$ and the singularity time scale $t_{c, u to 0}$ of the KPZ equation with an infinitesimal surface tension is investigated.
We study atypically large fluctuations of height $H$ in the 1+1-dimensional Kardar-Parisi-Zhang (KPZ) equation at long times $t$, when starting from a droplet initial condition. We derive exact large deviation function of height for $lambda H<0$, where $lambda$ is the nonlinearity coefficient of the KPZ equation. This large deviation function describes a crossover from the Tracy-Widom distribution tail at small $|H|/t$, which scales as $|H|^3/t$, to a different tail at large $|H|/t$, which scales as $|H|^{5/2}/t^{1/2}$. The latter tail exists at all times $t>0$. It was previously obtained in the framework of the optimal fluctuation method. It was also obtained at short times from exact representation of the complete height statistics. The crossover between the two tails, at long times, occurs at $|H|sim t$ as previously conjectured. Our analytical findings are supported by numerical evaluations using exact representation of the complete height statistics.
80 - Erwin Frey 1998
We study the scaling regimes for the Kardar-Parisi-Zhang equation with noise correlator R(q) ~ (1 + w q^{-2 rho}) in Fourier space, as a function of rho and the spatial dimension d. By means of a stochastic Cole-Hopf transformation, the critical and correction-to-scaling exponents at the roughening transition are determined to all orders in a (d - d_c) expansion. We also argue that there is a intriguing possibility that the rough phases above and below the lower critical dimension d_c = 2 (1 + rho) are genuinely different which could lead to a re-interpretation of results in the literature.
We use the optimal fluctuation method to evaluate the short-time probability distribution $mathcal{P}left(H,L,tright)$ of height at a single point, $H=hleft(x=0,tright)$, of the evolving Kardar-Parisi-Zhang (KPZ) interface $hleft(x,tright)$ on a ring of length $2L$. The process starts from a flat interface. At short times typical (small) height fluctuations are unaffected by the KPZ nonlinearity and belong to the Edwards-Wilkinson universality class. The nonlinearity, however, strongly affects the (asymmetric) tails of $mathcal{P}(H)$. At large $L/sqrt{t}$ the faster-decaying tail has a double structure: it is $L$-independent, $-lnmathcal{P}simleft|Hright|^{5/2}/t^{1/2}$, at intermediately large $|H|$, and $L$-dependent, $-lnmathcal{P}sim left|Hright|^{2}L/t$, at very large $|H|$. The transition between these two regimes is sharp and, in the large $L/sqrt{t}$ limit, behaves as a fractional-order phase transition. The transition point $H=H_{c}^{+}$ depends on $L/sqrt{t}$. At small $L/sqrt{t}$, the double structure of the faster tail disappears, and only the very large-$H$ tail, $-lnmathcal{P}sim left|Hright|^{2}L/t$, is observed. The slower-decaying tail does not show any $L$-dependence at large $L/sqrt{t}$, where it coincides with the slower tail of the GOE Tracy-Widom distribution. At small $L/sqrt{t}$ this tail also has a double structure. The transition between the two regimes occurs at a value of height $H=H_{c}^{-}$ which depends on $L/sqrt{t}$. At $L/sqrt{t} to 0$ the transition behaves as a mean-field-like second-order phase transition. At $|H|<|H_c^{-}|$ the slower tail behaves as $-lnmathcal{P}sim left|Hright|^{2}L/t$, whereas at $|H|>|H_c^{-}|$ it coincides with the slower tail of the GOE Tracy-Widom distribution.
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