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 Ap/As between the projected (frame) and the saturated areas. We show that thermal fluctuations modify the elastic energy of stretched surfaces (Ap/As>1), and dominate the elastic energy of compressed surfaces (Ap/As<1). When Ap approximately As the elastic energy is not much affected by the fluctuations; the frame area at which the surface tension vanishes becomes smaller than As and the area elasticity modulus increases.