Nonequilibrium steady state of a weakly-driven Kardar-Parisi-Zhang equation

Baruch Meerson, Pavel V. Sasorov, Arkady Vilenkin

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


We consider an infinite interface of 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 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 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 can be found from a minimization of the (nonlocal) equilibrium free energy of the interface. In contrast, the tails of 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 scales as In P(H) ∝ |H|, while the faster-decaying tail scales as In P(H) ∝ |H|3. The slower-decaying tail has important implications for the statistics of directed polymers in random potential.

Original languageAmerican English
Article number053201
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number5
StatePublished - 1 May 2018

Bibliographical note

Publisher Copyright:
© 2018 IOP Publishing Ltd and SISSA Medialab srl.


  • fluctuation phenomena
  • kinetic growth processes
  • large deviations in non-equilibrium systems
  • macroscopic fluctuation theory


Dive into the research topics of 'Nonequilibrium steady state of a weakly-driven Kardar-Parisi-Zhang equation'. Together they form a unique fingerprint.

Cite this