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Register a new userOptimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
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Atypically large fluctuations in macroscopic non-equilibrium systems continue to attract interest. Their probability can often be determined by the optimal fluctuation method (OFM). The OFM brings about a conditional variational problem, the solution of which describes the optimal path of the system which dominates the contribution of different stochastic paths to the desired statistics. The OFM proved efficient in evaluating the probabilities of rare events in a host of systems. However, theoretically predicted optimal paths were observed in stochastic simulations only in diffusive lattice gases, where the predicted optimal density patterns are either stationary, or travel with constant speed. Here we focus on the one-point height distribution of the paradigmatic Kardar-Parisi-Zhang interface. Here the optimal paths, corresponding to the distribution tails at short times, are intrinsically non-stationary and can be predicted analytically. Using the mapping to the directed polymer in a random potential at high temperature, we obtain snapshots of the optimal paths in Monte-Carlo simulations which probe the tails with an importance sampling algorithm. For each tail we observe a very narrow tube of height profiles around a single optimal path which agrees with the analytical prediction. The agreement holds even at long times, supporting earlier assertions of the validity of the OFM in the tails well beyond the weak-noise limit.
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Consider the short-time probability distribution $mathcal{P}(H,t)$ of the one-point interface height difference $h(x=0,tau=t)-h(x=0,tau=0)=H$ of the stationary interface $h(x,tau)$ described by the Kardar-Parisi-Zhang equation. It was previously shown that the optimal path -- the most probable history of the interface $h(x,tau)$ which dominates the higher tail of $mathcal{P}(H,t)$ -- is described by any of emph{two} ramp-like structures of $h(x,tau)$ traveling either to the left, or to the right. These two solutions emerge, at a critical value of $H$, via a spontaneous breaking of the mirror symmetry $xleftrightarrow-x$ of the optimal path, and this symmetry breaking is responsible for a second-order dynamical phase transition in the system. Here we employ a large-deviation Monte Carlo sampling algorithm in conjunction with the mapping between the KPZ interface and the directed polymer in a random potential at high temperature. This allows us to observe the optimal paths, which determine each of the two tails of $mathcal{P}(H,t)$, down to probability densities as small as $10^{-500}$. The observed mirror-symmetry-broken traveling optimal paths for the higher tail, and mirror-symmetric paths for the lower tail, are in good quantitative agreement with analytical predictions.
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 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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