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Register a new userScaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem
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Authors
Erwin Frey
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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.
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We study the dynamics of vortices in a two-dimensional, non-equilibrium system, described by the compact Kardar-Parisi-Zhang equation, after a sudden quench across the critical region. Our exact numerical solution of the phase-ordering kinetics shows that the unique interplay between non-equilibrium and the variable degree of spatial anisotropy leads to different critical regimes. We provide an analytical expression for the vortex evolution, based on scaling arguments, which is in agreement with the numerical results, and confirms the form of the interaction potential between vortices in this system.
Surface growth governed by the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than two undergoes a roughening transition from smooth to rough phases with increasing the nonlinearity. It is also known that the KPZ equation can be mapped onto quantum mechanics of attractive bosons with a contact interaction, where the roughening transition corresponds to a binding transition of two bosons with increasing the attraction. Such critical bosons in three dimensions actually exhibit the Efimov effect, where a three-boson coupling turns out to be relevant under the renormalization group so as to break the scale invariance down to a discrete one. On the basis of these facts linking the two distinct subjects in physics, we predict that the KPZ roughening transition in three dimensions shows either the discrete scale invariance or no intrinsic scale invariance.
Circular KPZ interfaces spreading radially in the plane have GUE Tracy-Widom (TW) height distribution (HD) and Airy$_2$ spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a cup, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as $langle L(t) rangle = L_0+omega t^{gamma}$, while their mean height $langle h rangle$ increases as usual [$langle h ranglesim t$]. We show that the competition between the $L$ enlargement and the correlation length ($xi simeq c t^{1/z}$) plays a key role in the asymptotic statistics of the interfaces. While systems with $gamma>1/z$ have HDs given by GUE and the interface width increasing as $w sim t^{beta}$, for $gamma<1/z$ the HDs are Gaussian, in a correlated regime where $w sim t^{alpha gamma}$. For the special case $gamma=1/z$, a continuous class of distributions exists, which interpolate between Gaussian (for small $omega/c$) and GUE (for $omega/c gg 1$). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for $omega/c approx 10$. Despite the GUE HDs for $gamma>1/z$, the spatial covariances present a strong dependence on the parameters $omega$ and $gamma$, agreeing with Airy$_2$ only for $omega gg 1$, for a given $gamma$, or when $gamma=1$, for a fixed $omega$. These results considerably generalize our knowledge on the 1D KPZ systems, unveiling the importance of the background space in their statistics.
We study the stochastically driven conserved Kardar-Parisi-Zhang (CKPZ) equation with quenched disorders. Short-ranged quenched disorders is found to be a relevant perturbation on the pure CKPZ equation at one dimension, and as a result, a new universality class different from pure CKPZ equation appears to emerge. At higher dimensions, quenched disorder turns out to be ineffective to influence the universal scaling. This results in the asymptotic long wavelength scaling to be given by the linear theory, a scenario identical with the pure CKPZ equation. For sufficiently long-ranged quenched disorders, the universal scaling is impacted by the quenched disorder even at higher dimensions.
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.
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