  Dilute or semi-dilute solutions of non-intersecting self-avoiding walk (SAW)
polymer chains are mapped onto a fluid of ``soft'' particles interacting via an
effective pair potential between their centers of mass. This mapping is
achieved by inverting the pair distribution function of the centers of mass of
the original polymer chains, using integral equation techniques from the theory
of simple fluids. The resulting effective pair potential is finite at all
distances, has a range of the order of the radius of gyration, and turns out to
be only moderately concentration-dependent. The dependence of the effective
potential on polymer length is analyzed in an effort to extract the scaling
limit. The effective potential is used to derive the osmotic equation of state,
which is compared to simulation data for the full SAW segment model, and to the
predictions of renormalization group calculations. A similar inversion
procedure is used to derive an effective wall-polymer potential from the center
of mass density profiles near the wall, obtained from simulations of the full
polymer segment model. The resulting wall-polymer potential turns out to depend
strongly on bulk polymer concentration when polymer-polymer correlations are
taken into account, leading to a considerable enhancement of the effective
repulsion with increasing concentration. The effective polymer-polymer and
wall-polymer potentials are combined to calculate the depletion interaction
induced by SAW polymers between two walls. The calculated depletion interaction
agrees well with the ``exact'' results from much more computer-intensive direct
simulation of the full polymer-segment model, and clearly illustrates the
inadequacy -- in the semi-dilute regime -- of the standard Asakura-Oosawa
approximation based on the assumption of non-interacting polymer coils.
