The effect of thermal fluctuations on Schulman area elasticity

We study the elastic properties of a two-dimensional fluctuating surface whose area density is allowed to deviate from its optimal (Schulman) value. The behavior of such a surface is determined by an interplay between the area-dependent elastic energy, the curvature elasticity, and the entropy. We identify three different elastic regimes depending on the ratio A _p/A_s between the projected (frame) and the saturated areas. We show that thermal fluctuations modify the elastic energy of stretched surfaces (A_p/A_s > 1), and dominate the elastic energy of compressed surfaces (A_p/A_s < 1). When A_p - A_s the elastic energy is not much affected by the fluctuations; the frame area at which the surface tension vanishes becomes smaller than A _s and the area elasticity modulus increases.