Statistical mechanics of membranes: freezing, undulations, and topology fluctuations

Membranes are two-dimensional sheets of molecules which are embedded and fl uctuate in three-dimensional space. The shape and out-of-plane fluctuations of tensionless membranes are controlled by their bending rigidity. Due to their out-of-plane fluctuations, flexible membranes exhibit very different behaviour to hat two-dimensional systems. We discuss three properties of me mbranes: (i) the renormalization of the bending rigidity in fluid membranes due to undulations on short length scales (ii) the suppression of the crys talline phase, and the hexatic-to-fluid transition; and (iii) the lamellar- to-sponge transition in Systems with variable topology. we focus on simulat ion studies, which are based on the numerical analysis of dynamically trian gulated surface models.