First-passage dynamics of linear stochastic interface models: numerical simulations and entropic repulsion effect

TL;DR

This paper investigates the first-passage dynamics of one-dimensional fluctuating interfaces using Langevin simulations, revealing how entropic repulsion affects the approach to a target height and assessing the accuracy of weak-noise theory.

Contribution

It provides a detailed numerical analysis of interface first-passage behavior, highlighting the impact of entropic repulsion on dynamic exponents and the limitations of weak-noise theory.

Findings

Average profile approaches target height algebraically over time.

Dynamic exponent is smaller than WNT prediction due to entropic repulsion.

Entropic repulsion analyzed via fractional Brownian motion model.

Abstract

A fluctuating interfacial profile in one dimension is studied via Langevin simulations of the Edwards-Wilkinson equation with non-conserved noise and the Mullins-Herring equation with conserved noise. The profile is subject to either periodic or Dirichlet (no-flux) boundary conditions. We determine the noise-driven time-evolution of the profile between an initially flat configuration and the instant at which the profile reaches a given height $M$ for the first time. The shape of the averaged profile agrees well with the prediction of weak-noise theory (WNT), which describes the most-likely trajectory to a fixed first-passage time. Furthermore, in agreement with WNT, on average the profile approaches the height $M$ algebraically in time, with an exponent that is essentially independent of the boundary conditions. However, the actual value of the dynamic exponent turns out to be significantly smaller than predicted by WNT. This "renormalization" of the exponent is explained in terms of the entropic repulsion exerted by the impenetrable boundary on the fluctuations of the profile around its most-likely path. The entropic repulsion mechanism is analyzed in detail for a single (fractional) Brownian walker, which describes the anomalous diffusion of a tagged monomer of the interface as it approaches the absorbing boundary. The present study sheds light on the accuracy and the limitations of the weak-noise approximation for the description of the full first-passage dynamics.