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Dynamical Scaling of Polymerized Membranes

Monte Carlo simulations have been performed to analyze the sub-diffusion dynamics of a tagged monomer in self-avoiding polymerized membranes in the flat phase. By decomposing the mean square displacement into the out-of-plane ($\parallel$) and the in-plane ($\perp$) components, we obtain good data collapse with two distinctive diffusion exponents $2 α_{\parallel} = 0.36 \pm 0.01$ and $2 α_{\perp} = 0.21 \pm 0.01$, and the roughness exponents $ζ_{\parallel} = 0.6 \pm 0.05$ and $ζ_{\perp} = 0.25 \pm 0.05 $, respectively for each component. Their values are consistent with the relation from the rotational symmetry. We derive the generalized Langevin equations to describe the sub-diffusional behaviors of a tagged monomer in the intermediate time regime where the collective effect of internal modes in the membrane dominate the dynamics to produce negative memory kernels with a power-law. We also briefly discuss how the long-range hydrodynamic interactions alter the exponents.