We study the bead-spring model for a polymerized phantom membrane in the overdamped limit, which is the two-dimensional generalization of the well-known Rouse model for polymers. We derive the exact eigenmodes of the membrane dynamics (the "Rouse modes"). This allows us to obtain exact analytical expressions for virtually any equilibrium or dynamical quantity for the membrane.As examples we determine the radius of gyration, the mean square displacement of a tagged bead, and the autocorrelation function of the difference vector between two tagged beads. Interestingly, even in the presence of tensile forces of any magnitude the Rouse modes remain the exact eigenmodes for the membrane. With stronger forces the membrane becomes essentially flat, and does not get the opportunity to intersect itself; in such a situation our analysis provides a useful and exactly soluble approach to the dynamics for a realistic model flat membrane under tension.PACS numbers: 02.50.Ey, 82.35.Lr
The dynamics of phantom bead-spring chains with the topology of a symmetric star with f arms and tadpoles (f = 3, a special case) is studied, in the overdamped limit. In the simplified case where the hydrodynamic radius of the central monomer is f times as heavy as the other beads, we determine their dynamical eigenmodes exactly, along the lines of the Rouse modes for linear bead-spring chains. These eigenmodes allow full analytical calculations of virtually any dynamical quantity. As examples we determine the radius of gyration, the mean square displacement of a tagged monomer, and, for star polymers, the autocorrelation function of the vector that spans from the center of the star to a bead on one of the arms.PACS numbers: 02.50.Ey, 82.35.Lr
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