Nonequilibrium Steady State of a Weakly-Driven Kardar-Parisi-Zhang Equation

Abstract

We consider an infinite interface in $d>2$ dimensions, governed by the Kardar-Parisi-Zhang (KPZ) equation with a weak Gaussian noise which is delta-correlated in time and has short-range spatial correlations. We study the probability distribution of the interface height $H$ at a point of the substrate, when the interface is initially flat. We show that, in a stark contrast with the KPZ equation in $d<2$, this distribution approaches a non-equilibrium steady state. The time of relaxation toward this state scales as the diffusion time over the correlation length of the noise. We study the steady-state distribution $\mathcal{P}(H)$ using the optimal-fluctuation method. The typical, small fluctuations of height are Gaussian. For these fluctuations the activation path of the system coincides with the time-reversed relaxation path, and the variance of $\mathcal{P}(H)$ can be found from a minimization of the (nonlocal) equilibrium free energy of the interface. In contrast, the tails of $\mathcal{P}(H)$ are nonequilibrium, non-Gaussian and strongly asymmetric. To determine them we calculate, analytically and numerically, the activation paths of the system, which are different from the time-reversed relaxation paths. We show that the slower-decaying tail of $\mathcal{P}(H)$ scales as $-\ln \mathcal{P}(H)\propto |H|$, while the faster-decaying tail scales as $-\ln \mathcal{P}(H)\propto |H{|}^3$. The slower-decaying tail has important implications for the statistics of directed polymers in random potential.

21 one-column pages, 9 figures, extended version

References in corpus (7)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• The one-dimensional KPZ equation and its universality class
• Optimal fluctuation approach to a directed polymer in a random medium
• Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension
• Universal and non-universal tails of distribution functions in the directed polymer and KPZ problems
• Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation
• Local average height distribution of fluctuating interfaces

Cited by in corpus (10)

• Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface
• Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line
• Field-Theoretic Thermodynamic Uncertainty Relation -- General formulation exemplified with the Kardar-Parisi-Zhang equation
• Observing symmetry-broken optimal paths of stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point
• Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• The Two Scaling Regimes of the Thermodynamic Uncertainty Relation for the KPZ-Equation
• Full distribution of the ground-state energy of potentials with weak disorder