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

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 the dependence of the relaxation time τ of a typical fluctuation governed by Eq. (1.1) or (1.2) on the system size L via τ ∝ L z , with z = 2 (EW equation), z = 4 (MH equation).(1.12)Large deviations of stochastic processes are formally described by Freidlin-Wentzel theory [35][36][37], which is equivalent to a Martin-Siggia-Rose/Janssen/de Dominicis path-integral formulation [38][39][40][41] in the limit of weak noise [42][43][44]. This approach provides an action functional, the minimization of which yields the most probable ("optimal") path connecting two states [e.g., Eqs. (1.5) and (1.4)]. For an explicit derivation of the corresponding weak-noise theory (WNT) for the EW and MH equation see, e.g., Refs. [30,45]. A related large deviation formalism in the context of lattice gases is reviewed in Ref. [46].An important predecessor to the present work is Ref. [30], where the WNT of Eq. (1.2) with periodic boundary conditions has been solved. Here, we extend that study by discussing further aspects of the first-passage dynamics, focusing, in particular, on the effect of boundary conditions. Within the WNT of Eqs. (1.1) and (1.2), we obtain minimum-action paths describing extremal fluctuations of the profile fulfilling Eqs. (1.4) and (1.5), without conditioning on the first-passage. We remark that the solution of WNT for Dirichlet no-flux boundary conditions [Eqs.( 1.7) and (1.8)] is technically involved since it requires the consideration of an adjoint eigenproblem [see Appendix B 1 a]. Predictions of WNT will be compared to Langevin simulations in an accompanying paper [47].The first-passage problem for the MH equation discussed here and in Ref.[47] is relevant, inter alia, for the rupture of liquid wetting films. In contrast to previous studies [9, 48-61], we focus here on the case where disjoining pressure is negligible and film rupture is solely driven by noise. A related WNT describing the noise-induced breakup of a liquid thread has been analyzed in Ref. [62]. Rare-event trajectories of the kind considered here are furthermore relevant fo...

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

“…The first-passage problem of the MH equation discussed here and in Ref. [1] is, inter alia, physically relevant for noise-driven rupture of liquid films on substrates. So far, typically films have been considered which are either linearly unstable with respect to small fluctuations of the interface or where the rupture proceeds via hole nucleation in the presence of disjoining pressure [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].…”

“…Within WNT, τ is in fact the characteristic time scale for the development of the first-passage profile in an equilibrium system (see Ref. [1]). This property is confirmed by the present simulations.…”

“…(1.1) and (1.2) has been discussed in detail in Ref. [1]. WNT represents a leading-order saddle-point approximation to the first-passage problem and describes the most-likely ("optimal") trajectory between two states.…”

“…Here and in Ref. [1], we focus on linearized models in one dimension and assume absence of any deterministic force beside surface tension. In particular, we neglect the influence of disjoining pressure, which is experimentally justified for colloidal fluids [35,36].…”

Dynamics of the critical Casimir force for a conserved order parameter after a critical quench

“…[80]), which is distinct from the fully conservative model B [39] (denoted as "model B B " in this context). A modification of Dirichlet BCs in order to implement the no-flux condition is possible but leads to transcendental eigenvalues [81,82] and is left for future investigations.…”