Divergent Fluctuations from an Infrared 2D-Mode Catastrophe

Abstract

Molecular simulations of interfacial polar media routinely employ periodic boundary conditions parallel to the interface. We show that this geometry injects a uniform plane mode ($q_{\parallel}=0$) that converts the plane-averaged electrostatic potential into a cumulative sum of plane-dipole increments, a random walk in $z$. Consequently, the variance of the plane-averaged potential grows linearly with depth in semi-infinite slabs and follows a parabolic Brownian-bridge profile in finite cells with both ends fixed, with an amplitude inversely proportional to the cell's lateral area. Hence, at any finite area, the variance diverges with slab thickness, a 2D-mode catastrophe. In contrast, a pure 1D chain (no lateral replication) and a fully 3D, nonperiodic medium both exhibit bounded fluctuations that saturate with distance. The mechanism is generic to any solver of Poisson's equation with 2D periodicity, so the apparent growth and ultimate divergence in potential fluctuation are artifacts of boundary conditions rather than material response, and we provide a simple scaling criterion for choosing slab sizes that keeps these artifacts under quantitative control.

Figures (1)

• Figure 1: Variance of the plane-averaged electrostatic potential under 2D periodicity. Comparing analytic theory with molecular-dynamics (MD) simulations for TIP3P water confirms the divergence of the variance in a 2D-periodic slab of a polar material of thickness 50 nm and lateral areas $A=\{1,4\}\,\mathrm{nm}^2$. (a) The variance increases proportional to $z$ away from a single electrode (random-walk behavior), and (b) follows $z(1-z/L)$ for confinement between two electrodes (Brownian-bridge behavior). The fitted slope $S$ scales as $1/A$, isolating the uniform plane mode ($q=0$), while a small constant offset $\sigma_0^2$ captures bounded contributions.