Measures of integration in calculating the effective rigidity of fluid surfaces

Abstract

Formulae differing by a numerical factor (1 vs. 3) have recently been obtained by several authors for the decrease of the bending rigidity of fluid surfaces caused by out-of-plane fluctuations. We compare the measures of integration (or the representations of the modes) for the example of the flexible rod, show why one-dimensional displacement measures are inadequate, and examine the curvature measure for surfaces. Our previous result (factor 1) is rederived in terms of coupling by geometry of energetically decoupled curvature fluctuation modes