Measure factors, tension, and correlations of fluid membranes

Abstract

We study two geometrical factors needed for the correct construction of statistical ensembles of surfaces. Such ensembles appear in the study of fluid bilayer membranes, though our results are more generally applicable. The naive functional measure over height fluctuations must be corrected by these factors in order to give correct, self-consistent formulas for the free energy and correlation functions of the height. While one of these correctionsthe Faddeev-Popov determinant -has been studied extensively, our derivation proceeds from very simple geometrical ideas, which we hope removes some of its mystery. The other factor is similar to the Liouville correction in string theory. Since our formulas differ from those of previous authors, we include some explicit calculations of the effective frame tension and two-point function to show that our version indeed secures coordinate-invariance and consistency to lowest nontrivial order in a temperature expansion.

Thermal fluctuations and bending rigidity of bilayer membranes

“…If we fit our simulation results to the form We turn now to the core of this question: whether the surface tension imposed by the barostat on a membrane γ 0 is equal or not to that estimated through the analysis of the microscopic fluctuations. Discussion about this problem can be found in previous works, which either suggests these surface tensions are equal [15][16][17] or different. [18][19][20] The barostat surface tension γ 0 is the conjugated variable to the projected area A 0 of the simulation box, while γ fluct , obtained from the membrane fluctuations, is related to the true area of the corrugated membrane A.…”

“…Advancing the discussion in Subsection IV B, we note that for the tensionless membranes a more sensitive representation of the low q fluctuations may be required to determine γ 0 . The expansions of (16) and (18) at low q imply that the bending modulus from the fit γ x (q) = γ o + κ x q 2 + · · · , depends on the representation of the undulatory mode,…”

“…The method relies on the measurement of the force needed to buckle a membrane. 14 In addition to the bending modulus equivalence, another open question is whether the surface tension obtained from the analysis of the thermal fluctuation spectrum of the flat membrane, agrees [15][16][17] or differs [18][19][20] from the externally imposed tension. This is a question we clarify in the present paper.…”

Flexoelectricity and thermal fluctuations of lipid bilayer membranes: Renormalization of flexoelectric, dielectric, and elastic properties

“…Two simplifications will be employed below: (1) the nonlinear Hamiltonian H 0 [v ′ ] is expanded and truncated with terms up to quadratics of v ′ , and (2) the nonlinear integration measure [Dv ′ ] is replaced by the "naive measure" [32]:…”

Theory of Vesicles and Droplet Type Microemulsions: Configurational Entropy, Size Distribution, and Measurable Properties

“…It is sufficient to note here that the calculation of z' from a model is far from trivial. For example, the problem of 'gauge fixing', which is required for such a calculation, is a very subtle matter, even for 'simple' on the average flat interfaces, as pointed out in [17].…”