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

Abstract

Two different tensions can be defined for a fluid membrane: the internal tension, <i>γ</i>, conjugated to the real membrane area in the Hamiltonian, and the frame tension, <i>τ</i>, conjugated to the projected (or frame) area. According to the standard statistical description of a membrane, the fluctuation spectrum is governed by <i>γ</i>. However, using rotational invariance arguments, several studies argued that the fluctuation spectrum must be governed by the frame tension <i>τ</i> 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 the arc length offers a natural parametrization, we give a new proof that the fluctuations are driven by <i>τ</i>, 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.