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 Delta F approximate to Ak(B)TL(-1)R(parallel to) for R-parallel to much greater than L. Here R-parallel to is the length of tube occupied by the chain, and A is a universal amplitude. We show how to determine Delta F and R-parallel to 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 approximate to LRparallel to-1[w(2)], where w is the winding number, are in excellent agreement with Monte Carlo simulations of Frauenkron, Cause, and Grassberger for strips with periodic boundaries. [S1063-651X(99)12105-6].