First-passage dynamics of linear stochastic interface models: weak-noise theory and influence of boundary conditions

TL;DR

This paper analyzes the first-passage dynamics of one-dimensional stochastic interface models, deriving universal scaling functions and examining the influence of boundary conditions on profile shapes in different regimes.

Contribution

It provides a weak-noise theoretical framework for the first-passage behavior of fluctuating interfaces under various boundary conditions, including explicit profile shape calculations.

Findings

Universal algebraic approach to maximum height attainment

Profile shape dependence on boundary conditions in equilibrium

Transient regime profile shape is boundary-condition independent

Abstract

We consider a one-dimensional fluctuating interfacial profile governed by the Edwards-Wilkinson or the stochastic Mullins-Herring equation for periodic, standard Dirichlet and Dirichlet no-flux boundary conditions. The minimum action path of an interfacial fluctuation conditioned to reach a given maximum height $M$ at a finite (first-passage) time $T$ is calculated within the weak-noise approximation. Dynamic and static scaling functions for the profile shape are obtained in the transient and the equilibrium regime, i.e., for first-passage times $T$ smaller or lager than the characteristic relaxation time, respectively. In both regimes, the profile approaches the maximum height $M$ with a universal algebraic time dependence characterized solely by the dynamic exponent of the model. It is shown that, in the equilibrium regime, the spatial shape of the profile depends sensitively on boundary conditions and conservation laws, but it is essentially independent of them in the transient regime.