Scaling of Moduli of Active and Thermal Elastic Membranes

Non-equilibrium and active effects in mesoscopic scale systems are of great interest for study and applications. We will discuss the role of these non-equilibrium active effects in the context of thermally fluctuating 2-D elastic membranes. We implement a generalization of the elastic tensor that includes odd elastic moduli that break both conservation of energy and angular momentum, due to Colin Scheibner et al. Breaking these symmetries means that deformations from a reference state can induce chiral forces that cannot be derived from a Hamiltonian. Thus, the behavior of odd elastic membranes must instead be investigated via non-equilibrium Langevin equations. These mechanical properties are obtained via calculation of the correlation functions of local displacements. Indeed, Nelson and Frey calculated these correlation functions for energy-conserving and fully permeable elastic membranes and showed that one can recoup the universal positive and negative power-law exponents of the bending rigidity and shear modulus respectively (eta, eta_u) (these results were obtained originally by Nelson and Peliti using a Boltzmann weight). Ultimately, we are interested in observing whether the introduction of odd moduli will change the scaling behavior of classical elastic membranes. When introducing the odd elastic modulus K (which couples pure shear and simple shear), we show that it reduces with exponent 2*eta_u, thus leaving the Aronovitz-Lubensky fixed point stable. When introducing the odd elastic modulus A (which couples dilation strains and torques), we find the behavior to be less trivial.