Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation

Naftali R. Smith, Baruch Meerson, Pavel Sasorov

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

We use the optimal fluctuation method to evaluate the short-time probability distribution P (H, L, t) of height at a single point, H = h (x = 0, t), of the evolving KardarParisiZhang (KPZ) interface h (x, t) 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 P(H). At large L/√t the faster-decaying tail has a double structure: it is L-independent, -lnP ∼ |H|5/2 /t1/2, at intermediately large |H|, and L-dependent, -lnP ∼ |H|2 L/t, at very large |H|. The transition between these two regimes is sharp and, in the large L/√t limit, behaves as a fractional-order phase transition. The transition point H = H+ c depends on L/√t. At small L/√t, the double structure of the faster tail disappears, and only the very large-H tail, -lnP ∼ |H|2 L/t, is observed. The slower-decaying tail does not show any L-dependence at large L/√t, where it coincides with the slower tail of the GOE TracyWidom distribution. At small L/√t this tail also has a double structure. The transition between the two regimes occurs at a value of height H = 1DUMMY-c which depends on L/√t. At L/√t → 0 the transition behaves as a mean-field-like second-order phase transition. At |H| < |1DUMMYc | the slower tail behaves as -lnP - |H|2 L/t, whereas at |H| > |Hc it coincides with the slower tail of the GOE TracyWidom distribution.

Original languageEnglish
Article number023202
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2018
Issue number2
DOIs
StatePublished - 8 Feb 2018
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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