Divergent Fluctuations from a 2D Infrared Catastrophe

TL;DR

This paper reveals how 2D periodic boundary conditions in molecular simulations cause artificial, diverging potential fluctuations due to unscreened in-plane modes, affecting the interpretation of interfacial polar media.

Contribution

It analytically characterizes the divergence caused by 2D periodicity and offers a criterion for choosing lateral cell dimensions to mitigate artifacts.

Findings

Potential variance grows linearly with depth in semi-infinite slabs.

In finite systems, potential fluctuations follow a parabolic profile.

Diverging fluctuations are artifacts of 2D periodic boundary conditions.

Abstract

Molecular simulations of interfacial polar media routinely employ periodic boundary conditions parallel to the interface. We show that this lateral periodicity introduces a spatially uniform in-plane mode ($q_{\parallel}=0$) that is unscreened because every lateral replica carries identical charge fluctuations. This 2D mode reduces the plane-averaged potential to a stochastic integral of the plane-averaged charge density along $z$, so that in a semi-infinite slab the variance of the potential grows linearly with depth. In a finite or periodic cell along $z$, with boundaries held at fixed potential, it follows a parabolic profile--a Brownian bridge--pinned to zero at both ends, with amplitude inversely proportional to the lateral cell area. These diverging fluctuations are a pure artifact of the imposed 2D lateral periodicity: they remain bounded in systems that are non-periodic or of finite lateral extent. We provide an analytic expression for their magnitude in dipolar media, yielding a practical criterion for the choice of lateral cell dimensions.