Nonequilibrium dynamics of a growing interface

A growing interface subject to noise is described by the Kardar-Parisi-Zhang equation or, equivalently, the noisy Burgers equation. In one dimension this equation is analyzed by means of a weak noise canonical phase space approach applied to the associated Fokker-Planck equation. The growth morphology is characterized by a gas of nonlinear soliton modes with superimposed linear diffusive modes. We also discuss the ensuing scaling properties.Comment: 14 pages, 11 figures, conference proceeding; a few corrections have been adde

Morphology and scaling in the noisy Burgers equation: Soliton approach to the strong coupling fixed point

The morphology and scaling properties of the noisy Burgers equation in one dimension are treated by means of a nonlinear soliton approach based on the Martin-Siggia-Rose technique. In a canonical formulation the strong coupling fixed point is accessed by means of a principle of least action in the asymptotic nonperturbative weak noise limit. The strong coupling scaling behaviour and the growth morphology are described by a gas of nonlinear soliton modes with a gapless dispersion law and a superposed gas of linear diffusive modes with a gap. The dynamic exponent is determined by the gapless soliton dispersion law, whereas the roughness exponent and a heuristic expression for the scaling function are given by the form factor in a spectral representation of the interface slope correlation function. The scaling function has the form of a Levy flight distribution.Comment: 5 pages, Revtex file, submitted to Phys. Rev. Let

Patterns in the Kardar-Parisi-Zhang equation

We review a recent asymptotic weak noise approach to the Kardar-Parisi-Zhang equation for the kinetic growth of an interface in higher dimensions. The weak noise approach provides a many body picture of a growing interface in terms of a network of localized growth modes. Scaling in 1d is associated with a gapless domain wall mode. The method also provides an independent argument for the existence of an upper critical dimension.Comment: 8 pages revtex, 4 eps figure

Canonical phase space approach to the noisy Burgers equation

Presenting a general phase approach to stochastic processes we analyze in particular the Fokker-Planck equation for the noisy Burgers equation and discuss the time dependent and stationary probability distributions. In one dimension we derive the long-time skew distribution approaching the symmetric stationary Gaussian distribution. In the short time regime we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer-replica model and asymmetric exclusion model results.Comment: 4 pages, Revtex file, submitted to Phys. Rev. Lett. a reference has been added and a few typos correcte

Weak noise approach to the logistic map

Using a nonperturbative weak noise approach we investigate the interference of noise and chaos in simple 1D maps. We replace the noise-driven 1D map by an area-preserving 2D map modelling the Poincare sections of a conserved dynamical system with unbounded energy manifolds. We analyze the properties of the 2D map and draw conclusions concerning the interference of noise on the nonlinear time evolution. We apply this technique to the standard period-doubling sequence in the logistic map. From the 2D area-preserving analogue we, in addition to the usual period-doubling sequence, obtain a series of period doubled cycles which are elliptic in nature. These cycles are spinning off the real axis at parameters values corresponding to the standard period doubling events.Comment: 22 pages in revtex and 8 figures in ep

Solitons and diffusive modes in the noiseless Burgers equation: Stability analysis

The noiseless Burgers equation in one spatial dimension is analyzed from the point of view of a diffusive evolution equation in terms of nonlinear soliton modes and linear diffusive modes. The transient evolution of the profile is interpreted as a gas of right hand solitons connected by ramp solutions with superposed linear diffusive modes. This picture is supported by a linear stability analysis of the soliton mode. The spectrum and phase shift of the diffusive modes are determined. In the presence of the soliton the diffusive modes develop a gap in the spectrum and are phase-shifted in accordance with Levinson's theorem. The spectrum also exhibits a zero-frequency translation or Goldstone mode associated with the broken translational symmetry.Comment: 9 pages, Revtex file, 5 figures, to be submitted to Phys. Rev.

Two-Loop Renormalization Group Analysis of the Burgers-Kardar-Parisi-Zhang Equation

A systematic analysis of the Burgers--Kardar--Parisi--Zhang equation in $d+1$ dimensions by dynamic renormalization group theory is described. The fixed points and exponents are calculated to two--loop order. We use the dimensional regularization scheme, carefully keeping the full $d$ dependence originating from the angular parts of the loop integrals. For dimensions less than $d_c=2$ we find a strong--coupling fixed point, which diverges at $d=2$, indicating that there is non--perturbative strong--coupling behavior for all $d \geq 2$. At $d=1$ our method yields the identical fixed point as in the one--loop approximation, and the two--loop contributions to the scaling functions are non--singular. For $d>2$ dimensions, there is no finite strong--coupling fixed point. In the framework of a $2+\epsilon$ expansion, we find the dynamic exponent corresponding to the unstable fixed point, which describes the non--equilibrium roughening transition, to be $z = 2 + {\cal O} (\epsilon^3)$, in agreement with a recent scaling argument by Doty and Kosterlitz. Similarly, our result for the correlation length exponent at the transition is $1/\nu = \epsilon + {\cal O} (\epsilon^3)$. For the smooth phase, some aspects of the crossover from Gaussian to critical behavior are discussed.Comment: 24 pages, written in LaTeX, 8 figures appended as postscript, EF/UCT--94/3, to be published in Phys. Rev. E

Canonical phase space approach to the noisy Burgers equation: Probability distributions

We present a canonical phase space approach to stochastic systems described by Langevin equations driven by white noise. Mapping the associated Fokker-Planck equation to a Hamilton-Jacobi equation in the nonperturbative weak noise limit we invoke a {\em principle of least action} for the determination of the probability distributions. We apply the scheme to the noisy Burgers and KPZ equations and discuss the time-dependent and stationary probability distributions. In one dimension we derive the long-time skew distribution approaching the symmetric stationary Gaussian distribution. In the short-time region we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer-replica and asymmetric exclusion model results. We also comment on the distribution in higher dimensions.Comment: 18 pages Revtex file, including 8 eps-figures, submitted to Phys. Rev.

Correlations, soliton modes, and non-Hermitian linear mode transmutation in the 1D noisy Burgers equation

Using the previously developed canonical phase space approach applied to the noisy Burgers equation in one dimension, we discuss in detail the growth morphology in terms of nonlinear soliton modes and superimposed linear modes. We moreover analyze the non-Hermitian character of the linear mode spectrum and the associated dynamical pinning and mode transmutation from diffusive to propagating behavior induced by the solitons. We discuss the anomalous diffusion of growth modes, switching and pathways, correlations in the multi-soliton sector, and in detail the correlations and scaling properties in the two-soliton sector.Comment: 50 pages, 15 figures, revtex4 fil