Frame tension governs the thermal fluctuations of a fluid membrane: new evidence

TL;DR

This paper demonstrates that the fluctuation spectrum of a fluid membrane is governed by the frame tension, $ au$, not the internal tension, $\gamma$, clarifying a longstanding debate through a new proof in a simplified one-dimensional model.

Contribution

The study provides a new proof that membrane fluctuations are governed by the frame tension and identifies the miscounting of configurations as the source of previous discrepancies.

Findings

Fluctuation spectrum is governed by frame tension $ au$.

Miscounting of configurations explains previous disagreements.

Analysis in a 1D model clarifies the tension governing fluctuations.

Abstract

Two different tensions can be defined for a fluid membrane: the internal tension, $\gamma$, conjugated to the real membrane area in the Hamiltonian, and the frame tension, $\tau$, conjugated to the projected (or frame) area. According to the standard statistical description of a membrane, the fluctuation spectrum is governed by $\gamma$. However, using rotational invariance arguments, several studies argued that fluctuation spectrum must be governed by the frame tension $\tau$ instead. These studies disagree on the origin of the result obtained with the standard description yet: either a miscounting of configurations, quantified with the integration measure, or the use of a quadratic approximation of the Helfrich Hamiltonian. Analyzing the simplest case of a one-dimensional membrane, for which arc length offers a natural parametrization, we give a new proof that the fluctuations are driven by $\tau$, and show that the origin of the issue with the standard description is a miscounting of membrane configurations. The origin itself of this miscounting depends on the thermodynamic ensemble in which calculations are made.