Divergent Fluctuations from a 2D Infrared Catastrophe

Richard G. Hennig; Clotilde S. Cucinotta

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arXiv · 2601.09009

Divergent Fluctuations from a 2D Infrared Catastrophe

Richard G. Hennig·Clotilde S. Cucinotta

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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.

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Richard G. Hennig, Clotilde S. Cucinotta. 2026-01-13. Divergent Fluctuations from a 2D Infrared Catastrophe. https://arxiv.org/abs/2601.09009

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