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Register a new userThe inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
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Added by
Alexandre Krajenbrink
Publication date
2021
fields
Physics
and research's language is
English
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We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing an approach which combines field theoretical, probabilistic and integrable techniques. We expand the program of the weak noise theory, which maps the large deviations onto a non-linear hydrodynamic problem, and unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system. Exact solutions, depending on the initial condition of the KPZ equation, are obtained using the inverse scattering method and a Fredholm determinant framework recently developed. These results, explicit in the case of the droplet geometry, open the path to obtain the complete large deviations for general initial conditions.
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We present the solution of the weak noise theory (WNT) for the Kardar-Parisi-Zhang equation in one dimension at short time for flat initial condition (IC). The non-linear hydrodynamic equations of the WNT are solved analytically through a connexion to the Zakharov-Shabat (ZS) system using its classical integrability. This approach is based on a recently developed Fredholm determinant framework previously applied to the droplet IC. The flat IC provides the case for a non-vanishing boundary condition of the ZS system and yields a richer solitonic structure comprising the appearance of multiple branches of the Lambert function. As a byproduct, we obtain the explicit solution of the WNT for the Brownian IC, which undergoes a dynamical phase transition. We elucidate its mechanism by showing that the related spontaneous breaking of the spatial symmetry arises from the interplay between two solitons with different rapidities.
The roughening of interfaces moving in inhomogeneous media is investigated by numerical integration of the phenomenological stochastic differential equation proposed by Kardar, Parisi, and Zhang [Phys. Rev. Lett. 56, 889, (1986)] with quenched noise (QKPZ). We express the evolution equations for the mean height and the roughness into two contributions: the local and the lateral one. We compare this two contributions with the ones obtained for two directed percolation deppining models (DPD): the Tang and Leschhorn model [Phys. Rev A 45, R8309 (1992)] and the Buldyrev et al. model [Phys. Rev. A 45, R8313 (1992)] by Braunstein al. [J. Phys. A 32, 1801 (1999); Phys. Rev. E 59, 4243 (1999)]. Even these models have being classified in the same universality class that the QKPZ the contributions to the growing mechanisms are quite different. The lateral contribution in the DPD models, leads to an increasing of the roughness near the criticality while in the QKPZ equation this contribution always flattens the roughness. These results suggest that the QKPZ equation does not describe properly the DPD models even when the exponents derived from this equation are similar to the one obtained from simulations of these models.
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.
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.
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.
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