Free energy of a long, flexible, self-avoiding polymer chain in a tube

The confinement free energy of a long, flexible, self-avoiding polymer chain, fluctuating in d spatial dimensions in an infinitely long cylindrical tube with diameter L, is given by DeltaF approximately Ak(B)TL(-1)R(parallel) for R(parallel)>>L. Here R(parallel) is the length of tube occupied by the chain, and A is a universal amplitude. We show how to determine DeltaF and R(parallel) from the correlation length xi(L)(t) of the n-vector model of magnetism in the limit n-->0, defined on the cylindrical volume, near the critical temperature t(c)(L) where xi(L) diverges. Using this correspondence, we estimate A in two dimensions from transfer-matrix data for self-avoiding walks on strips of width L with free, critically absorbing, and periodic boundaries. Our results for the universal amplitudes A and B approximately LR(-1)(parallel), where w is the winding number, are in excellent agreement with Monte Carlo simulations of Frauenkron, Causo, and Grassberger for strips with periodic boundaries.