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

Membranes are two-dimensional sheets of molecules which are embedded and fluctuate 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 flat two-dimensional systems. We discuss three properties of membranes: (i) the renormalization of the bending rigidity in fluid membranes due to undulations on short length scales; (ii) the suppression of the crystalline phase, and the hexatic-tofluid transition; and (iii) the lamellar-to-sponge transition in systems with variable topology. We focus on simulation studies, which are based on the numerical analysis of dynamically triangulated surface models.