On the weak scaling of the contact distance between two fluctuating interfaces with system size

TL;DR

This paper investigates the statistical behavior of the contact distance between two fluctuating interfaces evolving according to the Edwards-Wilkinson equation, revealing how this distance scales with system size and impacts interface annihilation.

Contribution

It introduces a general method to compute the distribution of contact distances and analyzes their scaling with system size for fluctuating interfaces.

Findings

The most likely contact distance scales with system size as a power or logarithmic function.

Simulations confirm the theoretical scaling predictions in two and three dimensions.

Topologically stable domains persist until the contact distance becomes very small, defying equilibrium expectations.

Abstract

A pair of flat parallel surfaces, each freely diffusing along the direction of their separation, will eventually come into contact. If the shapes of these surfaces also fluctuate, then contact will occur when their centers of mass remain separated by a nonzero distance $\ell$. Here we examine the statistics of $\ell$ at the time of first contact for surfaces that evolve in time according to the Edwards-Wilkinson equation. We present a general approach to calculate its probability distribution and determine how its most likely value $\ell^*$ depends on the surfaces' lateral size $L$. We are motivated by an interest in the motion of interfaces between two phases at conditions of thermodynamic coexistence, and in particular the annihilation of domain wall pairs under periodic boundary conditions. Computer simulations of this scenario verify the predicted scaling behavior in two and three dimensions. In the latter case, slow growth where $\ell^\ast$ is an algebraic function of $\log L$ implies that slab-shaped domains remain topologically intact until $\ell$ becomes very small, contradicting expectations from equilibrium thermodynamics.