Abstract
The statistical mechanical problem of two polymer molecules subject to a linking number constraint is solved in the so-called 'pre-averaged approximation', whereby a function of the coordinates of one of the molecules is replaced by its averaged value. For this molecule a critical winding number mc(L,s) can be identified, depending on the total length L of the molecules and the arc length scale s over which the configurational properties are being investigated. The authors find that mc approximately N3/4(L/s)1/4 and that for winding numbers m<mc(L,L) the original entropy of the chain still dominates and the configuration remains approximately Gaussian. For m>mc(L,L), the constraint dominates and evidence is presented to suggest that a partially collapsed solenoidal-like state is adopted, but one which is nevertheless still a Gaussian random coil on a smaller arc length scale s<s*, with s*=L(mc/m)4.